Project Risk Quantification with Monte Carlo Simulation

Open the Excel file named normal.xls. This file contains neither macros nor Monte Carlo simulation. We will demonstrate how Monte Carlo simulation is performed manually in Excel.

Let us recall that Monte Carlo simulation essentially combines two concepts:

1. The generation of random numbers.

2. Probability distributions.

Generating Random Numbers

We begin by generating a random number in Excel. To do this, we use the function RAND (in English). This is a volatile function, meaning it does not require arguments between parentheses. Every time we press F9 (recalculate), a continuous random number between 0 and 1 is generated. This allows us to create random numbers in Excel.

Next, let us create a probability distribution. The simplest and most universal probability distribution is the normal (Gaussian) distribution. This distribution requires two parameters:

1. Mean (300).

2. Standard deviation (100).

The standard deviation measures variability relative to the mean. With these numbers, we define or limit the range of visualization. A normal distribution theoretically spans from negative infinity to positive infinity. To limit this, we set the range between:

• Mean - 3 × standard deviation (as the minimum, 0 in this case).

• Mean + 3 × standard deviation (as the maximum, 600).

This allows us to generate random numbers between 0 and 600 for a normal distribution with a mean of 300 and a standard deviation of 100.

Calculating Probabilities

For example, let us consider the value 400. For this value, the normal distribution function in Excel (NORM.DIST) requires the following parameters:

• X: The value (in this case, 400).

• Mean: 300.

• Standard deviation: 100.

• Cumulative: True.

This indicates there is an 84% probability of generating a value up to 400 in a normal distribution with a mean of 300 and a standard deviation of 100. If we change the value to 300, the distribution’s symmetry means that 50% of the data will be below 300 and 50% will be above. Thus, 300 corresponds to the median or 50th percentile.

Now, let us generate a random number similar to the previous example using another Excel function: RANDBETWEEN. This function generates random integers between a minimum (0) and a maximum (600). Each time we press F9, it generates a random number within that range, which can be represented on a graph as a random point.

Cumulative Visualization

Each time we press F9, we generate a different random number, calculate its cumulative probability with NORM.DIST, and observe it on a graph. For instance:

• A random number of 65 has a cumulative probability of 1%.

• A random number of 340 accumulates a 66% probability.

This analysis can be extended by generating more random numbers. For example, using 5 points, each would have its corresponding probability. By increasing the number of points (10, 20, 100, or even 500), the cumulative probability curve becomes denser, offering a more detailed representation.

Finally, by generating 500 random data points, the complete curve is constructed cumulatively. Each time we press F9, Excel generates 500 random numbers distributed according to the "S" curve of the normal distribution.

Integrated Concepts

This is essentially Monte Carlo simulation, combining:

1. Random numbers.

2. Probability distributions.

This manual procedure provides a basic example of how Monte Carlo simulation works. Tools such as IziRisk Quantum, integrated into Excel as a ribbon, automate this process. IziRisk Quantum enables the generation of more complex simulations, managing multiple variables with probability distributions and random numbers efficiently.

This is Monte Carlo simulation, manually implemented in Excel.

The frequency-severity (or frequency-impact) model is a statistical framework used in operational and project risk management. This model was not invented by me; it originates in the insurance industry, where it is used to quantify and predict potential losses. It bases risk on two main components:

1. Frequency: Represents the number of loss events occurring within a defined period, typically a year. Frequency is modeled using discrete probability distributions, such as Bernoulli, Poisson, or Binomial distributions.

2. Severity or Impact: Represents the magnitude or size of the loss resulting from each event, usually expressed in monetary terms. Severity is modeled using distributions such as log-normal, Weibull, exponential, or Pareto.

The overall loss distribution is obtained by combining the frequency and severity distributions. This is done through a process called convolution, which we will explain in more detail later. This process sums the losses from individual events to produce the distribution of total or aggregated losses.

Steps of the Frequency-Severity Model

The model is developed in four main steps:

1. Define Frequency

Here, we estimate the number of events using historical data or expert judgment. We select an appropriate frequency distribution to model these events.

1. Model Severity or Impact

We fit a suitable distribution to the magnitude of the losses. This can be done using historical data or by relying on expert judgment, who can define a minimum, most probable, and maximum value for the severities of the risk being assessed.

1. Simulation

At this stage, we simulate aggregated losses. Monte Carlo simulation is used to combine the frequency and severity distributions through convolution. This estimates the distribution of total losses.

1. Quantify Risk Metrics

From the aggregated loss distribution, we calculate risk metrics such as Value at Risk (VaR) or Conditional Value at Risk (CVaR), which inform and support the decision-making process.

Practical Example: Workplace Accidents in a Project

Imagine a construction company wants to estimate potential losses from workplace accidents during a large project:

1. Frequency

The company reviews past projects and determines that, on average, eight workplace accidents occur per year in similar projects. Some projects recorded fewer than eight accidents, while others had more, but the average is eight. We use a Poisson distribution with a mean (λ) of eight to model the annual number of accidents, assuming these occur randomly and independently.

1. Severity or Impact

The company analyzes past incidents and finds associated costs (e.g., medical expenses, project delays, legal liabilities) ranging from $5,000 to $100,000. A log-normal distribution with a mean of $25,000 and a standard deviation of $15,000 is used to model these costs.

1. Monte Carlo Simulation

We generate the aggregated loss distribution by combining frequency and severity. Events are simulated using convolution to determine total loss costs. For each simulated accident, a cost is drawn from the log-normal severity distribution, and all losses for that year are summed.

1. Risk Metrics Analysis

For example, the Value at Risk (VaR) is calculated, representing the maximum estimated loss with a given confidence level, such as the 95th percentile. This means there is a 95% certainty that losses will not exceed this amount.

Conclusion
This is the essence of the frequency-severity model. It is a fundamental tool for understanding and structuring risk models in operational and project risk management. During this course, we will delve deeper into the practical applications of this model.

It is clear that to create a risk model, we will break the problem down into its two components: frequency and severity.

Now, we introduce the use of probability distributions. Remember, Monte Carlo simulation involves working with random numbers to construct probability distributions. From this perspective, probability distributions are divided into two main types:

1. Discrete Distributions

These generate only whole, finite numbers that can be specified as integers. They are used to model frequency—that is, to count how many events occur per unit of time (per year, month, etc.). Conventionally, we count events on an annual basis.

1. Continuous Distributions

These are used to model severity or impact—the monetary materialization of damages caused by risk events.

Discrete Distributions for Modeling Frequency

We focus first on discrete distributions, which are used to count the frequency of events.

By accessing the IziRisk platform in the cloud, navigate to the projects area and then to the risks area. Here, you can create a risk that will be divided into two components:

1. A frequency component.

2. An impact or severity component.

In the frequency section, four probability distributions are available:

1. Bernoulli Distribution

This is the simplest discrete distribution. It models events with only two possible outcomes (e.g., success or failure, occurs or does not occur).

1. Binomial Distribution

It models the number of successes in a fixed number of independent trials, where the probability of success remains constant in each trial.

1. Poisson Distribution

Ideal for modeling the number of events occurring in a fixed interval of time or space, assuming events occur randomly and at a constant rate.

1. Discrete Uniform Distribution

Assigns the same probability to all integer values within a specified range.

On the platform, there is a video tutorial for each of these distributions. We will analyze the characteristics of each distribution in detail to understand how to choose the most appropriate one for solving the first component of the risk model: the frequency problem—that is, how many events will occur per unit of time.

The simplest of the discrete distributions is the Bernoulli distribution. Let us begin with it.

The Bernoulli distribution requires only one parameter: p, the probability of success. This is a simple distribution because it generates only two possible outcomes: 0 or 1. That is, it is a binary distribution.

This distribution is typically used in scenarios where events can either occur or not occur—that is, they are binary. Therefore, its applicability is limited. Some typical cases include:

• Granting of a permit or patent (it is either granted or not).

• Discovery of a subsurface geological formation (it is either discovered or not).

When conducting risk matrix analyses, risks are often categorized using qualitative or subjective categories like "unlikely," "highly likely," etc. From my perspective, this categorization is inadequate because it naively assumes that the risk either occurs or does not occur. In reality, very few risks are strictly binary.

As mentioned earlier, binary events are specific cases, such as permits, concessions, patents, or discoveries in projects. Most risks are not binary.

For example, with a Bernoulli distribution where p=0.5p = 0.5p=0.5, there is a 50% probability that the event will not occur (0) and a 50% probability that it will occur (1). These are the only two possible outcomes.

If we change the parameter to p=0.2p = 0.2p=0.2, the probability that the event will not occur rises to 80% (0), while the probability of occurrence decreases to 20% (1). This parameter ppp can take any value between 0 and 1.

Alternatively, if we have an event with a probability of success p=0.95p = 0.95p=0.95, the event will not occur in 5% of cases (0), but it will occur in 95% of cases (1).

In summary, the Bernoulli distribution, the simplest of the discrete distributions, generates only zeros and ones. Its simplicity makes it ideal for modeling binary events, although its applicability to more complex risks is limited.

Think of the Binomial distribution as a sum or aggregation of Bernoulli distributions.

The Binomial distribution is discrete and generates integer numbers. It requires two parameters:

1. n: The number of samples exposed to a risk—that is, the number of elements or subjects exposed to that risk.

2. p: The homogeneous probability that the risk will occur.

A Binomial distribution with n=1 and p=0.5 is essentially a Bernoulli distribution with p=0.5. In this case, there is a 50% probability that the event will occur for that single element exposed to the risk and a 50% probability that it will not occur. This is a 50-50 scenario.

Example: A Single Machine

Suppose we have a machine that can fail within a year, with a failure probability of 10%. This means:

• In 90% of cases, there will be no failure (outcome 0).

• In 10% of cases, there will be a failure (outcome 1).

Example: Two Machines

What happens if we have two machines? This is where the parameter n (the number of elements exposed to the risk) comes into play. If n=2 and the probability of failure is p=0.1:

• In 81% of cases, there will be no failures (0 failures).

• In 18% of cases, there will be a single failure (1 failure).

• In 1% of cases, both machines will fail (2 failures).

Example: Ten Machines

If we increase n to 10 machines, all with a failure probability of p=0., we obtain a Binomial distribution with approximate results:

• A 35% probability of no failures.

• A 38% probability of exactly one failure.

• A 19% probability of exactly two failures.

Changing Probability p

If instead of p=0.1, we use p=0.25, the shape of the Binomial distribution changes. For example, with n=10:

• The scenario of two failures becomes the most probable (modal phenomenon) with a 28% probability.

• There is a 25% probability of exactly three failures.

• The probability of six failures among the ten machines is only 1.6%.

Practical Applications

The Binomial distribution is highly flexible and powerful when working with homogeneous groups of elements that share the same risk probability. Some examples include:

• Workplace Accidents: If you have a team of 20 workers, each with a 5% probability of having an accident, this can be modeled with a Binomial distribution.

• Project Suppliers: For 30 suppliers, each with a 20% probability of going bankrupt, the Binomial distribution helps calculate cumulative probabilities.

• ATM Network: If you manage 25 ATMs, each with a 5% probability of damage or theft, the Binomial distribution models the probabilities of various incidents.

Conclusion

The Binomial distribution can be thought of as an aggregated sum of Bernoulli distributions. It is a fundamental tool for risk management when dealing with multiple elements exposed to the same type of risk and with the same probability of occurrence.

Without a doubt, the most powerful probability distribution for modeling frequency in frequency-severity or frequency-impact models, whether in operational risks or project risks, is the Poisson distribution.

This distribution is both elegant and powerful because it assumes the following characteristics:

1. It requires only one parameter, λ (lambda):

Lambda represents the average or mean number of events that occur in a specific period.

1. It generates only positive integers:

This makes it particularly well-suited for modeling frequency-based events, such as the number of occurrences within a given timeframe.

Practical Example

Suppose we use the example of eight accidents per year. In this case, the parameter λ\lambdaλ is equal to 8, representing the mean or average number of events occurring annually.

The Poisson distribution generates positive integers: 0, 1, 2, and so on, theoretically extending to infinity. However, in practice, the probability of extremely large numbers decreases rapidly.

An interesting aspect of the Poisson distribution is that the value of λ\lambdaλ does not need to be an integer. For instance, if λ=2.5\lambda = 2.5λ=2.5, the distribution would yield probabilities such as:

• An 8% probability of no events occurring (0 events).

• A 20% probability of one event occurring (1 event).

• A 25% probability of two events occurring (2 events).

• A 21% probability of three events occurring (3 events).

The probabilities continue to diminish as the number of events increases, and for this example, events exceeding 10 would be highly unlikely.

Key Properties

1. Independence of Events:

The Poisson distribution assumes that events are independent of one another. This is crucial for modeling random occurrences over a fixed period or space.

1. Generates Integer Results:

While λ\lambdaλ can be a decimal, the resulting values are always integers, which is ideal for modeling frequency-based events like accidents, failures, or requests.

Summary
The Poisson distribution has the following fundamental characteristics:

• It requires only one parameter, λ\lambdaλ, which can be derived from historical data or provided by an expert’s judgment.

• λ\lambdaλ does not need to be an integer, but the results will always be whole numbers.

• It is ideal for modeling frequency-related events, such as counts per unit of time.

Thanks to these properties, the Poisson distribution is an excellent tool for modeling frequency in frequency-severity or frequency-impact models in operational or project risk management.

The Discrete Uniform Distribution is the most uncertain of all probability distributions. In fact, it is known as the distribution of maximum uncertainty.

All integer values between a specified minimum and maximum have the same probability of occurring. This distribution is used when there is high uncertainty about the values of a variable. It represents a state of maximum uncertainty when no single value is more likely than others, and an expert cannot specify a central or most probable value.

Practical Example

If we define a discrete uniform distribution between 0 and 100, every integer within that range has the same probability of occurring. The probability for each number is uniform, creating a rectangular shape in the probability graph.

If we adjust the range to 0 and 10, the same principle applies. All integers between 0 and 9 will have equal probabilities of occurring. In this case, with 10 possible outcomes, each has a probability of 10%.

Applications of the Discrete Uniform Distribution

This distribution is especially useful at the beginning of a project when we want to model variability or uncertainty but lack sufficient information to define a more precise central value or probability regarding a risk event.

Recommendation

While useful in early stages, it is not advisable to retain a discrete uniform distribution for final analyses. Over time, as more information becomes available, this distribution should be replaced with one that better reflects the uncertainty associated with the phenomenon being studied.

Continuous probability distributions are used to model the severity or impact in operational risks or project risks, specifically the monetary losses incurred from each risk event.

There are hundreds of continuous probability distributions, vastly expanding the range of options compared to discrete distributions. However, some distributions are far more commonly used than others.

The Normal Distribution, or Gaussian Distribution, is one of the most well-known. Many of us learned to use it in university statistics courses and often assume it is the default distribution. However, this is not necessarily true—the normal distribution is just one among many valid and useful options for modeling severity or impact.

Available Distributions for Modeling Severity

In the cloud-based application, several continuous probability distributions are available for modeling severity or monetary impact:

1. Uniform Distribution: Requires a minimum and a maximum value.

2. Exponential Distribution: Requires a single parameter, λ\lambdaλ.

3. Normal Distribution: Requires two parameters, μ\muμ (mean) and σ\sigmaσ (standard deviation).

4. Log-normal Distribution: Similar to the normal distribution but positively skewed.

These first four distributions are considered parametric because they require users to supply one or more parameters (μ\muμ, σ\sigmaσ, or λ\lambdaλ) to define their shape and range.

Empirical Distributions

In addition to parametric distributions, two empirical distributions are available:

1. Triangular Distribution

2. PERT Distribution: Popularized by the Project Management Institute (PMI) in its PMP certification.

These distributions are considered empirical because they do not require historical data for parameter calculation. Instead, they rely on expert judgment. Both distributions require three parameters:

• A minimum value.

• A most likely value.

• A maximum value.

With these three parameters, it is possible to construct distributions that reasonably reflect uncertainty when historical data is unavailable.

Advantages of Empirical Distributions

Empirical distributions are particularly useful in scenarios where explicit, historical data is unavailable for calculating parameters of parametric distributions. Instead of relying solely on data, we can use expert judgment to establish a minimum, most likely, and maximum value through a thoughtful and collaborative process.

This approach is far superior to qualitative or subjective risk assessments often found in risk matrices, such as “unlikely,” “low,” “high,” or “medium.” These categorizations tend to be abstract, subjective, and semantically weak.

By invoking quantitative criteria and building robust probability distributions, models become more precise and reliable. This allows for more effective incorporation of uncertainty into analyses, even in the absence of historical data.

Perhaps the most popular probability distribution is the Normal Distribution, also known as the Gaussian Bell Curve. Interestingly, it was not Gauss who originally formulated it but rather popularized it. The distribution was actually developed by a French mathematician named Abraham de Moivre.

The normal distribution has both ardent supporters and critics. Many of us likely recall our university statistics courses where this distribution was presented as a foundational probability tool. The normal distribution requires two parameters: μ\muμ (mu) and σ\sigmaσ (sigma), representing the mean and the standard deviation of a data sample, respectively.

Parameters of the Normal Distribution

1. The Mean (μ\muμ):

This is the simple average of all elements in a list.

1. The Standard Deviation (σ\sigmaσ):

This is a measure of variability that represents the average deviation of all points in a sample from the mean.

With these two data points—and assuming a sufficient dataset—we can construct a symmetric probability distribution. This symmetry means the central point (the mean) is equal to the mode and the median, and the data points are evenly distributed on either side of the perfectly symmetrical bell curve.

Limitations of the Normal Distribution

Real-world experiences show that most systems are not symmetric, making the normal distribution less applicable outside of theoretical examples. Although the normal distribution is based on the Central Limit Theorem—a foundational principle in probability—it is valid only under specific conditions.

In its simplest form, the Central Limit Theorem states that if we sum independent and random elements, their cumulative distribution will eventually converge to a normal distribution. While this is true for certain scenarios, such as opinion polls or surveys, many real-world systems deviate significantly from this pattern.

For instance, power-law distributions often feature asymmetric or heavy tails, meaning that the probabilities of extreme events are much higher than what a normal distribution would predict.

Critiques of the Normal Distribution

Among the notable critics of the normal distribution is Nassim Taleb, author of the bestseller The Black Swan. In his book, he dedicates a chapter titled The Great Intellectual Fraud to criticize the overreliance on the normal distribution. Taleb argues that events like the 2008-2009 financial crisis were partially caused by the misguided assumption that financial market variations followed a normal distribution. This crisis demonstrated that financial systems do not behave that way.

Issues with the Normal Distribution in Operational and Project Risks

For operational and project risks, the normal distribution is often unsuitable for two key reasons:

1. Dependence on Data:

Calculating the standard deviation requires historical data, which is often unavailable in project contexts. While the mean is relatively easy to determine, the standard deviation often requires a robust dataset.

1. Negative Values:

Consider a normal distribution with a mean of 1,000 and a standard deviation of 600. This configuration results in a significant probability of negative values, which is illogical in real-world risk scenarios. By nature, risks have a positive magnitude, beginning at zero. This makes the normal distribution an illogical choice for modeling monetary impacts in operational or project risks.

An Alternative: The Log-Normal Distribution

The log-normal distribution, often described as the close relative of the normal distribution, addresses many of the shortcomings of its counterpart. In the next chapter, we will explore the log-normal distribution and discuss how it effectively models monetary impacts in operational and project risks.

The Log-Normal Distribution is widely used to model severity or impact in operational and project risks within the context of the frequency-severity model. This distribution is particularly advantageous for several practical reasons:

Key Characteristics of the Log-Normal Distribution

1. Positive Values Only:

The log-normal distribution is defined solely for positive values, aligning naturally with operational losses since monetary losses cannot be negative.

1. Right-Skewed:

This distribution is skewed to the right, making it suitable for modeling loss events where most losses are small, but rare, high-severity events can occur.

1. Heavier Tails:

The log-normal distribution features heavier tails compared to the normal distribution, making it better suited for modeling extreme losses often observed in operational risks, such as fraud, natural disasters, system failures, or cybersecurity breaches.

1. Two Parameters (μ\muμ and σ\sigmaσ):

These parameters provide flexibility to adjust the distribution to various datasets:

1. • μ\muμ: Represents the median and reflects the central tendency in logarithmic space.

   • σ\sigmaσ: Represents the dispersion or variability of the data.

2. Ideal for Multiplicative Processes:

The log-normal distribution effectively models outcomes influenced by multiplicative factors, such as duration, scope, and magnitude of incidents.

1. Empirical Support:

Studies show that operational loss data often follows a pattern similar to the log-normal distribution, making it a natural choice for practical applications.

Integration with Frequency Models

The log-normal distribution pairs seamlessly with common frequency models like the Poisson distribution, allowing for efficient simulation of aggregated losses. Furthermore, its mathematical properties are well understood, simplifying the modeling and simulation of aggregated losses in statistical tools.

Versatility and Adoption

1. Adaptable to Different Scenarios:

The log-normal distribution is versatile enough to handle both low-frequency/high-severity and high-frequency/low-severity events.

1. Available in Tools:

Software like IziRisk offers tools to estimate log-normal parameters, making it easier to adopt this distribution for operational and project risk modeling.

1. Widely Recognized:

The log-normal distribution is broadly accepted in financial and operational risk management, making it reliable for regulatory reporting and stress testing.

Intuition Behind Parameters

• μ\muμ (mu): Represents the central tendency in logarithmic space.

• σ\sigmaσ (sigma): Captures the variability or volatility of losses.

Example of Building a Log-Normal Distribution

To construct a log-normal distribution, two parameters are needed:

1. A mean (μ\muμ), e.g., 100.

2. A standard deviation (σ\sigmaσ), e.g., 50.

With these values, a highly asymmetric curve is generated, accurately reflecting the nature of many operational risk events.

Conclusion
The log-normal distribution captures a wide range of operational losses, from small, frequent losses to catastrophic, rare ones. Its flexibility and robustness make it an ideal choice for modeling severity in operational and project risks.

The Exponential Distribution has numerous applications across various fields due to its unique and elegant properties. Below, we explore some of its most notable uses:

Applications of the Exponential Distribution

1. Modeling Time Between Events:

The exponential distribution is used to model the time between the occurrence of events in a process that follows a Poisson distribution. It can be thought of as the inverse of the Poisson distribution.

1. Reliability Analysis in Engineering:

The exponential distribution is employed to model the lifespan of components or systems with a constant failure rate. For example, it is used to predict when an electronic component might fail.

1. Risk and Finance:

In these fields, the exponential distribution is used to estimate the time until a risk event occurs, such as an accident or financial default.

1. Queue Theory:

In systems where events occur randomly and at a constant rate, the exponential distribution is used to model the arrival times of customers or service times.

1. Risk Analysis in Projects:

The exponential distribution is helpful for modeling the duration of certain risks or tasks with a constant probability of occurrence.

Example: Modeling Task Duration

Let’s consider historical data from 25 projects regarding the duration of a specific task. This data reflects the time, in days, that the task historically took to complete.

Analysis of this data reveals:

• It is highly asymmetric, skewed to the right.

• The majority of durations are concentrated between 0 and 1.8 days, with 11 out of 25 samples falling in this range.

• There are extreme values, with durations exceeding 14 days and reaching up to 17.5 days.

This type of data, which is strongly right-skewed, is effectively modeled using the exponential distribution.

Characteristics and Parameterization

The exponential distribution requires just one parameter, the mean (also known as beta). In this case, the average task duration from the historical data is calculated to be 4.15 days. Using this parameter, we can define an exponential distribution that accurately describes the skewed nature of the data and enables us to model the variability of task durations in future projects.

Practical Application in Software

In the IziRisk cloud application, the exponential distribution can be selected and parameterized with the value of 4.15 days to model task durations. This creates a curve that appropriately captures the variability and skewness of the task durations.

Advantages of the Exponential Distribution

1. Simplicity:
   It only requires a single parameter, making it easy to implement.

2. Adaptability:
   It captures asymmetric processes with a rightward skew, common in many risk management scenarios.

3. Efficiency with Limited Data:

The exponential distribution does not require extensive historical data to calculate or adjust.

Summary
The exponential distribution is a powerful, elegant tool for modeling asymmetric processes in project risk management and other disciplines.

The PERT Distribution, an acronym for Program Evaluation and Review Technique, is widely used in Monte Carlo simulation models. It is particularly useful in quantitative risk analysis for project management.

Why the PERT Distribution Is Valuable

This distribution is one of my personal favorites, especially when historical data is unavailable, and we must rely on the expertise of specialists. The PERT distribution is particularly suitable for modeling uncertainty in project-related variables, such as task durations, costs, or resource requirements.

Starting with the 6th Edition of the PMBOK® Guide (published in 2017), the use of the PERT distribution was more explicitly incorporated alongside the triangular distribution in project risk analysis. This inclusion marked a shift in risk management practices, recognizing the PERT distribution's ability to model uncertainties more realistically than the triangular distribution.

Advantages of the PERT Distribution

1. Smoothness and Realism:

Unlike the triangular distribution, which uses straight-line segments, the PERT distribution creates a smooth curve based on the Beta distribution. This results in a more realistic representation of uncertainties.

1. Three Key Parameters:

   • Optimistic Value: The best-case scenario, representing the minimum value.

   • Most Probable Value (Mode): The value most likely to occur.

   • Pessimistic Value: The worst-case scenario, representing the maximum value.

These three parameters reflect the variability experts observe in real-world scenarios.

1. Weighted Mean:

In the PERT distribution, the mean is calculated as a weighted average:

Mean=Optimistic+4×Most Probable+Pessimistic6\text{Mean} = \frac{\text{Optimistic} + 4 \times \text{Most Probable} + \text{Pessimistic}}{6}Mean=6Optimistic+4×Most Probable+Pessimistic​

1. Avoidance of Extreme Overestimations:

By emphasizing the most probable value, the PERT distribution reduces the likelihood of overestimating extreme scenarios.

Applications of the PERT Distribution

1. Monte Carlo Simulations:

Simulations using the PERT distribution produce reliable probability curves for assessing the likelihood of achieving project objectives.

1. Decision-Making Support:

By modeling expert inputs into probabilistic distributions, the PERT distribution helps stakeholders make informed decisions about project timelines, budgets, and risk mitigations.

1. Ease of Use and Interpretation:

With an intuitive three-point input system, the PERT distribution is easy for project managers and stakeholders to understand and use.

Beyond Project Management

Although commonly associated with project management, the PERT distribution is versatile and can be applied in:

• Finance

• Engineering

• Supply Chain Management

• Other fields requiring quantitative risk analysis

Using the PERT Distribution in IziRisk

In the IziRisk platform, the PERT distribution can be created by inputting the three required parameters (optimistic, most probable, and pessimistic). This allows for flexible, practical modeling of uncertainties.

Conclusion
The PERT distribution is a robust, intuitive, and flexible tool that enhances risk evaluation and mitigation, increasing the likelihood of project success when used in Monte Carlo simulations.

The Triangular Distribution is a straightforward and effective tool frequently used to quantify risks and uncertainties in projects. It is simple to understand and apply, as it only requires three parameters:

1. Minimum Value: The best-case scenario.

2. Most Probable Value (Mode): The most likely outcome.

3. Maximum Value: The worst-case scenario.

This distribution does not demand advanced statistical knowledge, making it accessible for effective use.

Characteristics of the Triangular Distribution

1. Suitability for Limited Data:

The triangular distribution is particularly useful when there is no historical data or detailed statistical inputs. Instead, it relies on expert judgment to define the three key parameters.

1. Flexibility:
   The triangular distribution can represent a wide range of shapes, depending on the relative positions of its three parameters. It can adapt to both symmetric and asymmetric distributions.

2. Ease of Visualization:

Its triangular shape is intuitive, making it easier for stakeholders to interpret quantified risks.

1. Statistical Properties:

It is simple to calculate key statistics, such as the mean and standard deviation, and it integrates seamlessly with Monte Carlo simulations for risk analysis.

Applications of the Triangular Distribution

1. PERT Technique:

The triangular distribution is often used in tools like PERT (Program Evaluation and Review Technique) to estimate task durations and costs, capturing:

1. • The best-case scenario (minimum).

   • The worst-case scenario (maximum).

   • The most likely outcome.

2. Project Risk Management:

The triangular distribution is applied to risks related to costs, timelines, and resource availability. It offers a concise and structured approach to profiling uncertainties.

1. Monte Carlo Simulation:

It serves as an ideal input for Monte Carlo simulations, helping to evaluate the cumulative impact of multiple risks on project objectives.

1. “What-If” Analysis:

It facilitates the evaluation of how changes in input values, such as optimistic versus pessimistic estimates, affect overall project risk.

1. Iterative Refinement:

As more data becomes available, estimates can be refined iteratively, improving the accuracy of risk evaluations over time.

Example
Consider a project task with uncertain duration:

• Minimum (Optimistic): 3 days.

• Most Probable: 5 days.

• Maximum (Pessimistic): 10 days.

The mean can be calculated as:

Mean=Minimum+Most Probable+Maximum3=3+5+103=6 days.\text{Mean} = \frac{\text{Minimum} + \text{Most Probable} + \text{Maximum}}{3} = \frac{3 + 5 + 10}{3} = 6 \text{ days}.Mean=3Minimum+Most Probable+Maximum​=33+5+10​=6 days.

In the IziRisk cloud application, these parameters can be articulated and visualized clearly, allowing project managers to make informed decisions about scheduling and resource allocation.

Summary
The triangular distribution balances simplicity and practicality, making it a valuable tool for managing project risks, especially for tasks with uncertain durations, costs, or outcomes.

This is a question frequently asked in project management: which distribution is better?

Both distributions use the same three parameters:

1. Minimum Value.

2. Most Probable Value (Mode).

3. Maximum Value.

However, while the input parameters are identical, the resulting probability curves differ significantly.

Key Differences Between PERT and Triangular Distributions

1. Triangular Distribution:

   • Has a triangular shape, defined by straight-line segments connecting the minimum, most probable, and maximum values.

   • Exhibits sharp corners at the minimum, maximum, and peak (most probable value).

2. PERT Distribution:

   • Features a smoother, more natural curve, as it is derived from the Beta distribution.

   • Transitions smoothly between the minimum and maximum values, with a gentler peak at the most probable value.

Impact of Asymmetry

The difference between PERT and triangular distributions becomes more pronounced in highly asymmetric scenarios.

For example, consider a distribution with the following parameters:

• Minimum: 100.

• Mode (Most Probable): 500.

• Maximum: 3,000.

When graphed:

• The triangular distribution assigns greater probability to extreme values on the right tail (e.g., values above 2,000).

• The PERT distribution, by contrast, assigns lower probability to extreme values, resulting in a curve that appears more realistic for many natural processes.

This makes the triangular distribution more conservative, as it assigns greater weight to extreme outcomes.

Why Choose PERT?

1. Continuity:

The PERT distribution avoids the sharp corners present in the triangular distribution, resulting in a smoother and more mathematically continuous curve.

1. Natural Shape:

The PERT distribution reflects the natural behavior of most processes, with gradual transitions and fewer abrupt changes.

1. Software Automation:

Although triangular calculations (e.g., mean, variance) are simpler, modern software tools like IziRisk handle PERT calculations automatically, making this complexity irrelevant to users.

Recommendation
For scenarios with significant asymmetry, the PERT distribution is recommended over the triangular distribution due to its smoother shape and better representation of natural processes. In scenarios with minimal asymmetry, the differences between the two distributions are negligible.

In a Continuous Uniform Distribution, all numbers within a defined range have an equal probability of occurrence. It can be likened to a bingo or lottery system where every number has an equal chance of being drawn.

Discrete vs. Continuous Uniform Distributions

1. Discrete Uniform Distribution:

Generates integer values within a specified range (e.g., 1, 2, 3, …, 10).

1. Continuous Uniform Distribution:

Generates continuous values, including decimals, within a specified range. For example:

1. • If the range is 0 to 9, numbers like 4.5 or 7.2 have the same probability as 3 or 6.

   • If the range is 0 to 1, values are probabilistic, meaning any fraction within the range has equal likelihood.

Applications of the Uniform Distribution

The uniform distribution is particularly useful as a baseline for introducing maximum uncertainty in variables where little or no information is available. It is often employed at the start of projects to model variability in simulations, especially when no prior data exists.

However, as more data becomes available, it is advisable to replace the uniform distribution with a more precise distribution to better reflect the actual uncertainty of the variable in question.

Applications
In the next chapter, we will explore advanced applications of the uniform distribution, including:

Creating probability segments.

• Building kernel distributions.

• Aggregating multiple distributions to construct creative and composite models.

The world of probability distributions is infinite. When asked how many probability distributions exist, the answer is limitless. While specialized software often provides access to dozens or even hundreds of predefined distributions, the practical focus is typically on a select few that are particularly popular and versatile.

Custom Distributions

There is a creative branch of analysis where you can design custom probability distributions tailored to specific needs. Consider the following example:

Cybersecurity Risk Example

Imagine you are modeling a cybersecurity risk, where extreme events are challenging to predict but could have a significant impact. Let’s assume the risk can be divided into three categories or segments of damage:

1. Low Scenario:

   • Frequency: 60% of events fall into this category.

   • Impact: Damages range from 0 to 100 as the most likely value, and up to 500 as the maximum value.

2. Medium Scenario:

   • Frequency: 30% of events are categorized as medium.

   • Impact: Damages range from 200 to 2,000, with 1,000 as the most likely value.

3. High Scenario:

   • Frequency: Only 10% of events are considered high impact.

   • Impact: Damages range from 1,500 to 10,000, with 3,000 as the most likely value.

Creating the Distribution in Excel

Using Excel, we can construct a system that generates random scenarios using a uniform distribution:

1. Generate Random Numbers:

In cell B1, generate a random number between 0 and 1 using the RAND function.

1. • If the value is less than 0.6, assign it to the low category.

   • If it’s between 0.6 and 0.9, assign it to the medium category.

   • If it’s above 0.9, assign it to the high category.

2. Assign Categories:

Use formulas in the C column to determine the selected category based on the random number.

1. Calculate Impact:

In cell B9, use a SUMPRODUCT function to calculate the monetary impact of each scenario, considering the probabilities and impacts defined for each category.

Monte Carlo Simulation

By running this simulation over 50,000 iterations, we can generate a probability curve that summarizes all possible risk scenarios.

Log-Normal Approximation

After completing the simulation, the results can be summarized using a log-normal distribution, which captures the following:

• Frequent small events: Represented by the majority of the data concentrated on the lower end.

• Rare extreme events: Represented by a long tail extending toward higher impacts.

1. Parameter Calculation:

   • Mean: Derived from the simulation results.

   • Standard Deviation: Calculated from the variability in the results.

2. Synthesis:
   Use these parameters to generate a log-normal distribution that approximates the behavior of the simulated data.

3. Comparison:
   Compare the custom distribution with the synthesized log-normal distribution. In most cases, both distributions will closely align in terms of their mean, standard deviation, and percentiles.

Conclusion

This example demonstrates how a creative and structured approach can transform a complex scenario into a manageable model. By leveraging custom distributions and tools like Excel or IziRisk, it becomes possible to build advanced risk models that integrate multiple layers of probability.

Remember that in the frequency-severity model (or frequency-impact), the process requires a fusion or integration of these two elements. This means breaking down a risk into:

1. Frequency: How often it occurs per unit of time (e.g., per month, per year).

2. Impact or severity: The monetary impact of each event.

Once these elements are atomized, they must be integrated again. Here, a common mistake inherited from the qualitative risk domain often occurs: simply multiplying frequency by severity.

Issues with Direct Multiplication

Multiplying frequency by severity is correct only when applied to the mean. However, it does not yield accurate values at the extremes of the resulting distribution. This can lead to overestimating the size of the tails, resulting in undesirable consequences, such as incorrectly calculating probabilistic contingencies.

For example, if a probabilistic contingency is set at the 80th or 90th percentile and it is miscalculated, excessive contingency values might be assumed.

Three Approaches to Integrate Frequency and Impact

We will proceed to integrate frequency and impact using three different approaches.

1. Frequency:

We define a Poisson distribution with an average (λ or lambda) of 4. The probabilistic graph of this distribution shows that the most frequent events are 3 or 4. For example:

• The probability of 8 events occurring is approximately 3-4%.

• For 9 or 10 events, the probability decreases progressively to nearly zero.

2. Impact or Severity:

We use a log-normal distribution, which requires two parameters:

• Mean (μ): 100.

• Standard deviation (σ): 150.

This creates a highly asymmetric distribution, where the mean is surpassed by the standard deviation, resulting in a strong concentration of low values but with a long tail extending to higher values.

First Approach: Direct Multiplication

We multiply the cell containing frequency by the cell containing impact. This approach is simple but problematic, as it exaggerates the tails, especially when frequency and impact simultaneously generate high values.

Second Approach: One-to-One Generation

For this approach:

1. We use the Poisson distribution to generate the number of events (frequency).

2. For each generated event, we independently assign a monetary impact using the log-normal distribution.

For example:

• If 8 events are generated, each will have a different random impact according to the log-normal distribution.

• If no event occurs, the impact is 0.

Afterward, all the generated impacts are summed for each iteration. This approach accurately captures variability in the tails.

Comparison Between the Two Approaches

After running 10,000 iterations and comparing the resulting curves:

• Mean: Both curves (multiplicative and one-to-one) tend to be similar in terms of the mean.

• 95th Percentile: Differences emerge here, with the multiplicative curve showing more exaggerated tails.

• Maximum: At this point, the multiplicative method can generate values up to double those produced by the one-to-one approach.

This demonstrates that the multiplicative method, by assuming events with high frequency and simultaneously high magnitudes, exaggerates values in the tails.

Third Approach: Convolution with Specific Functions

To avoid errors, we use a convolution function. This function:

1. Automatically generates frequency values using the Poisson distribution (λ = 4).

2. Integrates impact values using the log-normal distribution (mean = 100, standard deviation = 150).

In tools such as Easy Quantum for Excel, this convolution function is built-in and automates the process.

Results of the Convolution Approach

After running the simulation with this third approach and comparing it with the other two:

• Mean: Matches the previous approaches.

• 95th Percentile: Shows significant differences from the multiplicative approach, aligning more closely with the one-to-one method.

• Maximum: The convolution approach avoids exaggerating the tails, delivering more realistic results.

Conclusion

The correct approach to integrate frequency and impact is through proper convolution, which respects the probabilistic nature of both components. While direct multiplication is simple, it exaggerates the tails and produces unrealistic results.

In advanced tools like Easy Quantum for Excel, this convolution function is readily available, allowing for efficient and precise integration using a Poisson distribution for frequency and a log-normal distribution for impact.

This module introduces the concept of the Gantt chart. Some may find this module very basic, so its study is optional. However, at the end of this module, we include a section on incorporating uncertainties, which makes it different from what we usually encounter.

A Gantt chart is a powerful project management tool that visually represents a project’s timeline. It shows tasks, their duration, and the sequence in which they need to be completed. This chart helps track progress, allocate resources, and ensure that a project stays on schedule. Below is an introduction on how to create, use, and interpret a Gantt chart; and finally, how to turn it from a deterministic Gantt chart to a probabilistic one.

1. CREATING A GANTT CHART

• List Tasks: Start by listing all the tasks or activities that need to be completed in the project. This includes the major phases and subtasks.

• Define Duration: Estimate how long each task will take. This can be measured in days, weeks, or months, depending on the project's scope.

• Establish Dependencies: Identify any tasks that cannot begin until others are completed. For example, Task B may depend on Task A being finished before it can start.

• Create the Schedule: Plot the tasks along a horizontal axis, with time progressing from left to right. Each task is represented by a bar that spans the duration of the task, showing when it starts and ends.

• Assign Resources (optional): You can also include the resources (e.g., team members or equipment) responsible for each task, which helps in resource allocation and management.

2. USING A GANTT CHART

• Track Progress: As the project progresses, you can update the Gantt chart by marking tasks that are completed or indicating the percentage of progress for ongoing tasks.

• Adjust Schedules: If there are delays or changes, the Gantt chart can be modified to reflect new start and end dates, helping to maintain an up-to-date view of the project.

• Manage Dependencies: Use the chart to monitor dependencies between tasks and ensure they are completed in the proper sequence.

• Resource Allocation: A Gantt chart can highlight periods of overwork or underutilization of resources, helping to adjust and ensure efficient use of manpower and tools.

3. INTERPRETING A GANTT CHART

• Bars Represent Tasks: Each horizontal bar represents a specific task. The length of the bar shows the duration of the task, while the position indicates its start and end dates.

• Dependencies Between Tasks: Arrows or lines may appear connecting tasks to show dependencies. For example, an arrow from Task A to Task B indicates that Task B cannot start until Task A is completed.

• Milestones: Important project milestones are often represented as diamond shapes on the chart. These milestones mark significant achievements or deadlines within the project.

• Progress Indicators: Some Gantt charts include progress bars within the task bars to show the current status of each task (e.g., 50% completed).

A Gantt chart is an excellent tool for visualizing the timeline and workflow of a project. It helps project managers plan, schedule, and track task completion while identifying potential delays and resource allocation issues. By regularly updating and interpreting a Gantt chart, you can ensure that the project stays on track and meets its deadlines.

4. CONSIDERING UNCERTAINTIES

Uncertainty in the duration of activities is a common factor in most projects, as many tasks depend on variables that cannot always be predicted precisely; that is, they contain an element of uncertainty. To address this uncertainty, Gantt charts can include time ranges or probability estimates instead of a fixed duration for each activity. This allows for a more realistic management of possible variations in task execution time.

For this reason, note that we have inserted three columns in this Gantt chart to make it probabilistic, i.e., non-deterministic. This means we will consider that tasks or activities could have levels of uncertainty in terms of their duration. An expert or group of experts assigns three values: minimum, most likely, and maximum duration for each task in the schedule.

The iziRisk project management tool allows for probability-based estimates. This is achieved using time distributions, such as the triangular distribution or PERT, which we have previously described. An optimistic, pessimistic, and most likely time can be assigned to each task. In this way, Monte Carlo simulations will be generated to forecast future scenarios.

5. ADVANTAGES OF CONSIDERING UNCERTAINTY

• Realism: By representing the uncertainty in the duration of tasks, the schedule becomes more realistic, as it reflects the difficulties and unforeseen events that may arise during the project execution.

• Better Decision-Making: Time ranges and probability estimates allow project managers to make more informed decisions about resource allocation and contingency planning.

• Risk Identification: Incorporating uncertainty helps identify and visualize areas of the project that are more prone to delays, allowing for prioritization in risk management.

• Greater Flexibility: Uncertainty margins provide greater flexibility in scheduling, allowing for adjustments when needed.

Incorporating uncertainty into the duration of activities in a Gantt chart is an effective way to reflect the real conditions of the project, enabling better planning and risk management. While it presents certain disadvantages, such as added complexity and the need for specialized tools, the benefits of greater realism and flexibility in project management can outweigh these drawbacks. By integrating these elements of uncertainty, project teams can be better prepared to face unforeseen events and adjust their plans accordingly.

To introduce uncertainty into the duration of project activities, the first step is to define the level of uncertainty associated with each one.

In the parameters sheet, different levels of variation can be conventionally established. For example, medium uncertainty levels typically range from a minimum of 15% to a maximum of 25%. This means that when using probability distributions like PERT or triangular, the percentage of variation relative to the most likely value (modal point) must be defined. In cell B8, the type of distribution to be used is selected.

Thus, variability ranges can be set as a percentage of the estimated duration of each task. For example, if Task 13: Development has a most likely duration of 20 days, a variation of -5% on the lower end and +10% on the upper end can be assigned, resulting in values between 19 and 22 days, depending on the chosen distribution.

In cell F15, where the probability distribution function for the duration of Task 13 (Development) is generated, we will keep only the PERT probability distribution. When enabling the distribution chart, observe how the distribution looks while variability remains low.

Now, let's change the variability of this task to Medium. This will adjust the minimum and maximum values to 17 and 25 days, reflecting 15% variation on the lower side and 25% on the upper side. Notice how, when the chart is regenerated, the probability curve expands.

When the Monte Carlo simulation is executed, the randomly generated values for each activity’s duration will be aligned with these parameters. If the level of uncertainty changes from low to high, the range of variation will also adjust. For instance, a duration initially set at 20 days could now range between 14 days (minimum) and 30 days (maximum), reflecting greater uncertainty.

We can modify the function to use a Triangular distribution instead. When visualizing it, we will see that the shape of the distribution changes, but the parameters for minimum, most likely, and maximum values remain the same.

Remember, in the Parameters sheet, we can switch between PERT and Triangular distributions. Later in the course, we will analyze whether this change affects the final simulation results.

Now, let’s restore the original function in column F by copying it from the upper cell. This function includes a few additional elements that we will explore later in the course.

Summary

This approach allows us to model uncertainty more accurately in project planning, activity by activity. The key is to determine, based on expert judgment, whether each activity inherently has low, medium, or high uncertainty. Of course, these variability levels can be adjusted for each category as needed.

By incorporating uncertainty into the planning process, this method provides a more realistic foundation for decision-making.

In project management, the distinction between fixed and variable costs refers to how these costs behave concerning the project's progress and its activities.

Fixed Costs in a Project

Fixed costs are those that remain constant regardless of the duration of the task to which they are associated. These costs are incurred independently of how long the activity takes.

Examples of fixed costs in project activities:

• Rental of offices or temporary work sites

• Salaries of project administrative staff

• Software licenses required for project execution

• Depreciation of machinery or equipment

• Costs of permits or certifications

Variable Costs in a Project

Variable costs are those that depend on the duration of the associated task. In other words, they increase or decrease based on how long the activity lasts.

Examples of variable costs in project activities:

• Wages for labor hired on an hourly or daily basis

• Energy consumption in construction or production equipment

Distinguishing Fixed and Variable Costs from Direct and Indirect Costs

It is important to clarify that the distinction between fixed and variable costs is not the same as the difference between direct and indirect costs in a project.

Direct Costs

Direct costs are those that can be assigned directly to a specific activity within the project.

Indirect Costs

Indirect costs are those that cannot be attributed to a specific activity but are necessary for the overall operation of the project.

Examples of indirect costs in a project:

• Administrative and project management costs

• Utilities for the project office

• General maintenance and security expenses

Examples of direct costs in a project:

• Materials specifically used in a project task

• Wages paid to workers performing a specific task

However, direct costs can be either fixed or variable, depending on how they are structured.

Examples of Cost Classification in a Project

Labor Cost Example:

If workers are paid for the time they spend on a specific task, and the task is delayed, their wages are still a direct cost because they are linked to that task. However, the cost becomes variable because it depends on the task's duration.

Equipment Rental Example:

If a piece of equipment is rented exclusively for a construction phase:

• It is a direct cost because it is linked to a specific construction activity.

• It is variable if the rental is charged per day of use, meaning that if the activity is delayed due to a risk, the total cost will increase accordingly.

• It is fixed if the rental is paid as a flat fee regardless of the number of days used.

Key Takeaway: Focus on Duration-Based Costs

For project planning and cost analysis, the most relevant distinction is between fixed and variable costs based on task duration.

How to classify costs in a project?

• If a cost increases based on the duration of the task, it is variable.

• If a cost remains constant regardless of how long the task takes, it is fixed.

Why Does This Matter?

Ultimately, we want to analyze how task durations and costs fluctuate together under the impact of risks. Risks and uncertainties often have a direct impact on task duration, which in turn affects costs.

By correctly linking task duration and associated costs to potential risks, we ensure a realistic integration of three critical elements: risks, duration, and costs. This approach provides a better understanding of project behavior and allows for more accurate financial and scheduling projections.

1. Introduction

When managing a project, it is essential to understand how to insert fixed costs and variable costs into the planning. To do this, we use a structure based on activities, where each cost is directly associated with a specific task.

In our spreadsheet, column B references the activities and allows us to structure the costs as follows:

• Fixed costs

• Variable costs

• Total cost sum

2. Cost Analysis in Deterministic Parameters

Before considering risks, we can perform a deterministic analysis in the cost sheet. Here, we observe that:

• The sum of fixed costs is 12,000

• The sum of variable costs is 11,000

• The total project cost amounts to 23,000

If we take a specific task as an example, such as Task 13 - Development, we must assign it a probable fixed cost value in the corresponding column.

2.1. Defining Fixed Costs

The fixed cost is assigned regardless of the number of activity days. In the case of Task 13, a probable fixed cost of 2,000 has been defined.

Additionally, just as we assigned levels of uncertainty (low, medium, or high) to tasks, we apply the same principle to fixed and variable costs.

In column C, we can add variability to the fixed cost. For example, if the uncertainty is low, the fixed cost of 2,000 could fluctuate between:

• 1,900 (minimum, with a -5% variation)

• 2,200 (maximum, with a +10% variation)

If the uncertainty is medium, the range expands from 1,700 to 2,500, and if it is high, it extends from 1,400 to 3,000.

2.2. Defining Variable Costs

Variable costs are calculated per activity day. In column H, we can assign different levels of uncertainty and define:

• Minimum value

• Most probable value

• Maximum value

Since variable cost is charged per day, we must consider the number of days for each activity. In column M, we find the number of days assigned to each task.

The total variable cost of an activity is calculated using the following formula:

Total Variable Cost=Variable Cost per Day×Number of DaysTotal \ Variable \ Cost = Variable \ Cost \ per \ Day \times Number \ of \ DaysTotal Variable Cost=Variable Cost per Day×Number of Days

This calculation varies depending on the chosen level of uncertainty.

3. Impact of Changes in Calculation Parameters

When we change the parameters in B7 from deterministic to probabilistic, the values change dynamically. Pressing F9 updates:

• The fixed cost (column G)

• The variable cost per day (column L)

• The number of activity days (column M)

This process allows us to see how activity duration influences variable costs, reflected in column N (Total Variable Cost). Finally, in column O, we obtain the total project cost, integrating both fixed and variable costs.

4. Conclusion

Project cost calculation depends on a proper association between activities, fixed costs, and variable costs. Integrating uncertainty into these values allows for more realistic and flexible project management projections.

1. Introduction to Risks

We now move on to the risk section, which serves as the linking element between tasks and costs, both of which we have already explored.

In the "risks" sheet, we find a table that, for the purposes of this course, has been pre-filled with ten different types of risks.

• Risks are detailed in column C.

• In column B, risks can be included or excluded for marginal analysis. This allows us to evaluate scenarios such as:

  • What would happen if I did not include a specific risk?

  • How would my timeline look if a particular risk were fully mitigated?

2. Types of Risks

Risks can be classified into two major categories:

1. Activity-Associated Risks

• Directly impact specific tasks.

• Have a two-dimensional impact:

  • Frequency: How often the risk occurs.

  • Impact: Affects both the duration of the associated task and its cost.

• Example: Risks 1, 2, 7, and 8 belong to this category.

2. Cross-Cutting Risks

• Affect the entire project and cannot be linked to a specific task.

• Do not impact duration, as they are not tied to a particular activity.

• Only impact cost.

3. Measuring Risk Impact

3.1. Monetary Impact

We measure the monetary impact using three values found in the following columns:

• P → Minimum value

• Q → Most likely value

• R → Maximum value

The S column uses a PERT or triangular distribution to generate random values based on these parameters.

Example:
The risk "Low-quality code or insufficient testing" has:

• Minimum cost: $500

• Most likely cost: $1,000

• Maximum cost: $2,000

The model will generate random numbers within this range in cell S3.

3.2. Risk Frequency Measurement

We also measure the frequency of the risk, meaning how often it may occur per unit of time.

There are three probability distributions we can use to quantify risk frequency:

1. Bernoulli Distribution

• Example: The first risk has a 20% probability of occurring.

• In cell J3, the frequency will be 1 in 20% of cases and 0 in the remaining 80%.

• This type of risk is binary: it either happens or it doesn’t.

2. Binomial Distribution

• Example: Risk 2 ("external dependencies and supplier delays") affects task 14.

• It assumes there are 8 external suppliers, each with a 10% probability of experiencing a delay.

• Cell J4 will show the number of delays, which could range from 0 to 8, with decreasing probability as the number increases.

4. Impact on Duration and Cost

If a risk materializes, it can impact both the duration and cost of a task.

4.1. Impact on Duration

Example:
If Risk 2 (external dependencies and supplier delays) occurs:

• Minimum duration increase: 2 days

• Most likely increase: 5 days

• Maximum increase: 8 days

Each occurrence of the risk leads to a delay within this range.

4.2. Impact on Cost

In addition to affecting duration, this risk also generates an additional cost:

• Minimum cost: $400

• Most likely cost: $500

• Maximum cost: $1,000

Important:
The cost of the risk is independent of the cost caused by task delays.

5. Differentiating Between Risk and Uncertainty

We must remember that there are uncertainties related to task duration, which should not be confused with risks.

Example:

• Task 14 has costs that vary based on the number of days.

• The number 15, in this case, represents an actualized duration but could have been another value.

• This should not be confused with the duration impact caused by risks when they materialize.

5.1. Differentiating Risks in the Task Sheet

In the "Tasks" sheet, there is a column that tracks additional risk-related days for activities linked to specific risks.

Example:

• Tasks 13, 14, and 16 have associated risks that may increase their duration.

• These additional days are accounted for in the risk model.

On the other hand:

• The uncertainties in columns H, I, and J reflect other variations in duration that are not tied to a specific risk.

Key Takeaways

• Activity risks cause specific delays when they materialize.

• General uncertainties cause variations in duration without being linked to a specific risk.

6. Using the Risks Sheet

The "RISKS" sheet provides additional details about each risk.

• In cell A2, we can enter the risk number to view detailed information about its nature.

• This sheet is for informational purposes only and does not affect model calculations.

7. Conclusion and Next Steps

In this section, we have created an initial risk table, including:

1. Risk details

2. Frequency and probability distribution

3. Impact on duration and cost

In the next part of the course, we will explore mitigation strategies, analyzing which risks can and should be mitigated, and how this affects the risk table.

As we saw in Section 2 on Monte Carlo simulation, where we introduced some probability distributions, and as we will review with examples in this section, the fundamental approach to quantitatively assessing risks in a project is by analyzing them through their two essential components:

1. Risk Frequency: Represents how many times a risk event occurs over a period of time. It is generally expressed in non-negative integer values (e.g., the number of times a system failure occurs in a month).

2. Severity or Impact: Represents the monetary cost associated with the materialization of a risk.

Probability Distributions in Risk Simulation

To model these two components, we will use different probability distributions:

Distributions to model risk frequency:

• Bernoulli: For events that either occur or do not occur (e.g., a system failure).

• Binomial: For events that occur a finite number of times within a set of trials (e.g., supplier delivery delays).

• Poisson: For events that repeat over a period of time or space (e.g., the number of machine breakdowns per month).

• Discrete Uniform: For events where all possible values have the same probability within a defined range.

Distributions to model the monetary impact of risk:

• PERT: Ideal for modeling cost uncertainty with a minimum, most likely, and maximum value.

• Triangular: Similar to PERT but simpler to calculate and assign probabilities.

Integrating Frequency and Impact

To combine both elements (frequency and impact) in the simulation, a mathematical process called convolution is used, allowing us to calculate the total effect of a risk on the project.  It is important to note that this process is not simply a multiplication.

Note: This process was explained in Section 2.17: Convolution of Frequency and Severity.

In this section, we will delve deeper into how to model and understand project risks using these concepts.

As we have seen, the Bernoulli distribution is a binary distribution that allows only two states: zero or one. The risk either occurs or does not occur. Let’s look at the following example: we have a risk of damage to the ecosystem. This is a risk that either exists or does not exist, it materializes or does not materialize. In this case, it is a transversal risk, meaning it is a risk that can affect a variety of tasks or, rather, is not associated with a particular task.

This risk will then be quantified in terms of its frequency using a Bernoulli distribution. We will assign a parameter of 0.25, meaning there is a 25% chance that, in one out of every four possible cases, the risk will materialize. And if it does materialize, it will have no impact on the duration of any task, as it is not even associated with a task in this particular case. However, it does have a cost.

Let’s assume that the cost of this risk is a minimum of 1,000, with a most probable value of 2,000 and a maximum value of 5,000. With these two variables, we can then outline a specific case of ecosystem damage as a binary risk. Other types of risks that fall within this category could be, for example, patents or permit concessions, where the risk either occurs or does not occur.

In this case, we model it with a Bernoulli distribution, which could be viewed in the cloud application as follows: it is a risk of ecosystem damage with a description stating that it is a general risk, not associated with a task. It is a risk that will be categorized in some way using a Bernoulli distribution with a probability of 0.25.

We can then see in the graph that the frequency distribution is divided into two: there will be a 75% probability that the risk does not materialize (i.e., is zero), and a 25% probability that the risk will materialize (i.e., is one).

On the other hand, in terms of cost, this could materialize with a PERT distribution, with a minimum value of 1,000, a most probable value of 2,000, and a maximum value of 5,000. The risk application has the ability to combine both the frequency component using the Bernoulli distribution and the impact or severity component using the PERT distribution. This way, it creates the convoluted distribution of how this risk would appear once we run it within the simulation.

Let’s now think about another risk, a sum of Bernoullis. This is known as a binomial distribution. Imagine that the risk could materialize or not, but for a set of homogeneous elements, for example, always within the category of environmental risks.

Imagine that, in the case of our mine, there are 12 permits to be requested—12 environmental permits with regulatory requirements. This is also a transversal risk. We can model it with a binomial distribution. That is, we can take the first parameter of a binomial, which is n (the number of elements), in this case, 12, because there are 12 regulatory processes that require the same probability of occurrence. The probability in this case is 20%, meaning there is a 20% chance that the risk will materialize, that any of those 12 regulatory requirements will be denied.

This is as if we were doing it together, instead of having 12 Bernoulli distributions, we combine them and make a binomial. This binomial would then be articulated as follows: as a binomial with n = 12 and a probability of 20%. In the cloud application, what we can see in this case is that we create a binomial distribution with n = 12 and a 20% probability. This then articulates a histogram with the different scenarios that could materialize. With each of these probabilities, the amounts of impact, cost, and monetary severity would be generated, which in this case would have a monetary impact with a minimum of 1,000, a most probable value of 3,000, and a maximum value of 10,000.

By combining both elements, frequency with a binomial distribution (n = 12, p = 20%) and a PERT distribution for the impact and severity, we obtain the convoluted distribution of both impacts of this particular risk.

Another example of the binomial distribution in contractual risks is related to conflicts or even insolvencies of suppliers or contractors. Imagine that in a project, we have 35 suppliers or contractors. If we assume that each one of them has the same probability of having conflicts, say 20%, we can create a binomial distribution with n = 35 (the number of contractors) and p = 20%. This will generate a probability distribution that will determine the number of conflicts that could occur probabilistically.

Another example is the insolvency of suppliers or contractors. Suppose these same 35 contractors each have a 2% probability of going into insolvency. Let’s imagine that, in the history of this project or this project-based company, one out of every 50 contractors eventually goes bankrupt. One in 50 means a 2% probability. If we have 35 contractors, the binomial distribution would model the number of contractors that might go bankrupt if these events occurred with a binomial n = 35 and p = 2%.

As mentioned earlier, perhaps the most powerful and elegant of the probability distributions for frequency is the Poisson distribution. The Poisson distribution, as a reminder, only requires one parameter: lambda, which is the mean frequency. In other words, how many times per unit of time (per year) a phenomenon occurs.

For example, imagine that in this project, there is a history where the average number of natural damage events (storms, floods, etc.) is three per project. Using this parameter, lambda = 3, we could generate a Poisson distribution that would articulate a frequency distribution, showing the probability of encountering each of these scenarios.

Another example could be conflicts with the local community. If there is historical data or at least the judgment of an expert, they might say that, on average, they would expect to have about five conflicts. It’s not necessarily going to be five; it could be three, five, or ten. But what the Poisson distribution does is that by using a single criterion, lambda = 5, we articulate a probability distribution that spreads various probabilities, which, when simulated many times, will result in an average of five.

This allows us, with just one parameter, to articulate a frequency distribution that enables us to generate the frequency. That is, how many times per year or how many times during the duration of the project a phenomenon like natural events, conflicts with the local community, or any other type of risk event frequency would occur. This can be calculated either with historical information or the judgment of an expert, using an average value. We take that average value, that’s the lambda, we input it into a Poisson distribution, and through that, we can articulate very powerful frequencies for risk events.

In the cloud application, you just need to select the Poisson distribution for frequency, input the corresponding lambda parameter (for example, five community problems per year in the project), and obtain a probability distribution of how many events we would expect.

For example, in this case, we would expect that, given a lambda of 5, there is a 0% probability that no events will occur, a 3% probability that one event will occur, an 8% probability that two events will occur, a 17% probability for four events, and a 17% probability for five events. Given a lambda of five events per unit of time, the probability of having ten events or even more starts to decrease, dropping to less than 2%.

The uniform integer distribution is a discrete probability distribution in which all integer values within a specific range have the same probability of occurring. It is known as the maximum uncertainty distribution.

It is used in Monte Carlo simulations when there is no preference for one value over another within the specified range.

In risk analysis, the uniform integer distribution is applied when there is insufficient information to assign different probabilities to different events.

Example

If a project team estimates that a delivery delay could be between 3 and 7 days, and they assume that any value within this range is equally probable, the duration of the delay follows a uniform integer distribution with a minimum of 3 and a maximum of 7. Only integer numbers are generated.

Key Applications

The uniform integer distribution is a useful tool when all integer values within a given range are equally probable. It applies to situations where there is no additional information to favor some values over others and is commonly used in: Monte Carlo simulations, games of chance, uncertainty modeling and random selection processes.

Use it as a starting point to introduce variability into models. However, as more information becomes available, it is generally advisable to replace it with another distribution that contains more refined data.

In project risk management, estimating the potential impact of risks is crucial for effective decision-making. Two widely used probability distributions for modeling uncertainty in risk impacts are the PERT distribution and the Triangular distribution. Both rely on three key parameters:

• Minimum value (a) – The best-case scenario.

• Most likely value (b) – The most probable outcome.

• Maximum value (c) – The worst-case scenario.

However, they differ in how they weigh the probability of different outcomes, influencing their suitability for different project scenarios.

The Program Evaluation and Review Technique (PERT) distribution is a continuous probability distribution derived from the Beta distribution. It is commonly used when there is a need for smoother probability weighting, where the most likely estimate is given greater influence over the final outcomes.

The formula for calculating the mean or expected value on a PERT assigns 4 times more weight to the most likely value (b), making the PERT distribution a better choice when the best estimate is considered reliable.

A project team estimates that a construction cost overrun may range from $50,000 to $150,000, with the most likely outcome being $100,000. This result leans toward the most likely outcome, making it a suitable approach when the middle estimate is well-founded.

On the other hand, the Triangular distribution is a simpler probability distribution that forms a triangle-shaped probability curve. It does not assign extra weight to the most likely value, meaning the probability of values is more evenly distributed across the range.

Unlike PERT, the Triangular distribution assumes equal importance for all three parameters, making it more suitable when uncertainty is high, and the most likely estimate is not necessarily more reliable.

A project manager estimates that a software project delay could be between 10 and 30 days, with the most likely delay being 20 days. Using the Triangular distribution, the expected delay is:

Since the Triangular distribution does not give extra weight to the most likely value, this approach is more suitable when there is no clear reason to favor a specific estimate.

• Use the PERT distribution when you have high confidence in the most likely estimate and want to reduce the influence of extreme values.

• Use the Triangular distribution when uncertainty is higher, and all three estimates should be considered equally.

Both distributions are valuable tools in Monte Carlo simulations and risk assessment models, helping project managers make data-driven decisions. We will eventually compare in this course the outcomes produced by using these two distributions. We will focus on whether there is a significant difference in outcomes produced by one distribution or the other one.

Second Part of the Course: From Deterministic Models to Monte Carlo Simulation

Welcome to the second part of the course! Up to this point, we have built the fundamental concepts needed to understand the simulation methodology.

What we have learned so far:

Basic concepts of Monte Carlo simulation and probability distributions.

The three key components of the project model: tasks, costs, and risks.

Now, our goal is to apply this methodology to assess how to meet project objectives, ensuring that it is completed on time and within budget. In this section, we will go step by step through solving the example model.

1. Initial Deterministic Evaluation

We will begin in cell B7 of the Parameters sheet by setting up a deterministic model.

What is a deterministic model in Monte Carlo?

A deterministic model is one that does not incorporate randomness in its calculations or results. This means that:

· It always produces the same result if the input data remains the same.

· There is no variability in the parameters: probability distributions are not used.

· Multiple scenarios are not generated because uncertainty is not considered.

Example: A project with a deterministic model

Let's assume that the project starts on March 1, 2026 (cell D2 in the Tasks sheet).

Estimated results without uncertainty:

Completion date: July 25, 2026 (cell E2).

Project duration: 146 days (cell F1).

Estimated costs:

• Fixed costs: $12,000 (cell G1).

• Variable costs: $11,000 (cell N1).

• Total cost: $23,000 (cell O1).

Important: In this model, we have NOT yet introduced variability in duration and cost estimates.

Recording deterministic values:

Let’s enter the following values in the Parameters sheet:

C5: Deterministic completion date: July 25, 2026.

C6: Deterministic total cost: $23,000.

As we progress with new exercises, we will compare this deterministic scenario with more complex results based on simulation.

One major limitation of the deterministic model is that it does not allow for scenario evaluation or probability calculation. To address this limitation, we will explore two different approaches:

Approach 1: Traditional Scenario Analysis

Scenario analysis is a technique where a limited number of predefined scenarios are evaluated.

The three most common scenarios are:

· Optimistic → Best possible conditions.

· Most likely → Base estimate.

· Pessimistic → Worst possible conditions.

Problem with traditional scenario analysis:

· It only considers three specific scenarios, ignoring many other possible combinations.

· It does not capture all project uncertainties.

· It does not allow probability calculations to determine the likelihood of each outcome.

Approach 2: Monte Carlo Simulation (A More Robust Alternative)

Monte Carlo simulation is a more advanced methodology that evaluates thousands of possible scenarios by using probability distributions and computational power.

Key advantages of Monte Carlo over traditional scenario analysis:

· Evaluates multiple combinations of uncertainty, rather than just three scenarios.

· Uses probability distributions instead of fixed values.

· Captures the real variability of the problem.

· Allows probability calculations, such as the likelihood of exceeding a cost or deadline.

What will we do next?

In the upcoming exercises, we will apply Monte Carlo simulation to calculate probabilistic reserves and assess the impact of uncertainty on project outcomes. ?

3. Conclusion: Why Is This Change in Approach Important?

· The deterministic model is limited → It does not account for uncertainty.

· Traditional scenario analysis is useful but incomplete → It only evaluates a few scenarios.

· Monte Carlo simulation is the best option → It allows for a more complete assessment of risk and uncertainty.

Throughout this section, we will compare the results obtained with both approaches and see how Monte Carlo simulation helps us make more informed and precise project management decisions.

Let’s move on to the next module!

6.2. Definition of Output Variables in Monte Carlo Simulation

Once a Monte Carlo simulation model has been structured with its cause-and-effect relationships, meaning that input variables influence output variables, we can proceed with the definition of output variables.

In the context of a project duration and cost model, the output variables are those that allow us to evaluate these two fundamental dimensions. In the Outputs sheet, we find a section from B2 to B8, which contains seven cells with the variables we want to analyze and record within the model.

Once the simulation is executed, we will analyze each of these variables in detail:

Duration Dimension

1. Completion date → Defined in relation to cell E2 in the Tasks sheet, which contains the project’s completion date.

2.  Total duration in days → Found in cell F1 of the Tasks sheet.
3.  Total risk-related days → Located in cell K1 of the Tasks sheet, representing the additional days caused by the materialization of risks affecting project duration.

Cost Dimension

4. Total fixed cost of activities → Recorded in cell G1 of the Costs sheet.

5.  Variable cost of activities → Found in cell N1 of the Costs sheet.

6. Total cost of risks → The sum of the values in column T, from T3 to T12, representing the impact of risks in terms of costs.

7. Total project cost → Obtained by summing:

• Fixed cost of activities.

• Variable cost of activities.

• Total cost of risks.

With these seven variables defined, we now have the key indicators to analyze and control our project.

Configuring Output Variables in the Simulation

For the simulation software to monitor these variables, we must configure them correctly:

1. Select the range in the Outputs sheet, from B2 to B8.

2. Use the “Output Data” button, which will automatically assign the variable names based on the titles.

3. Optionally, we can rename these variables by clicking on each one.

From this point forward, these seven cells will be declared as output variables within the simulation model.

Verification and Execution of the Simulation

To confirm that the variables have been correctly configured:

-  Click on the third button of “Show Inputs and Outputs” within the software.
- Verify that the variables have been correctly declared as output variables.
-  During the next simulation, the system will populate these cells with the data generated in each iteration.

With this configuration, the model is now ready to run multiple iterations and evaluate the results obtained for each of these variables.

Final Summary

- We identified the key output variables in duration and costs.

- We configured the software to record these values during the simulation.

- We verified that the variables were correctly declared.

- We prepared the model to run simulations and analyze the results.

This process ensures that our Monte Carlo simulation model is ready to provide accurate and useful quantitative analysis for project management.

In the creation, analysis, and interpretation of the results of a Monte Carlo simulation, it is fundamental to define the output variables beforehand. While input variables may be optional, their presence allows us to establish a cause-and-effect relationship with the output variables.

In other words, having input variables helps determine which ones—or what set of variables—have the most significant impact on one or more output variables.

Relationship Between Input and Output Variables

A clear example is the total cost of a project. We know that this output variable is influenced by three components:

- Fixed cost

- Variable cost of activities

- Total cost of risks

All this information is found in the Costs sheet, where column O summarizes:

- Fixed cost

- Variable cost

- Risk-related cost

If we examine the range O3 to O30, we can see 28 input variables, each corresponding to a specific activity. To analyze them in the simulation context, we can declare them as input variables.

Configuring Input Variables

To define these input variables in the simulation software, follow these steps:

1.  Select the data range in column O.

2.  Use the "Input Data" button to declare these variables.

3. Assign appropriate names:

• By default, the software will take the names from the leftmost text in the same row.

• If these names are not suitable (for instance, if they come from column H), we can reference the correct range.

• 4. Use the "Select Names" button to assign names from B3 to B30, which contain the relevant information.

• 5. Press "Add Input" to register these variables in the simulation model.

To verify the correct configuration, we can use the third button in the "Show Inputs and Outputs" icon, where the declaration of the newly added variables will be displayed.

Running the Simulation and Analyzing the Impact

Now that the variables are defined, we can proceed with the simulation. The software will generate multiple iterations, linking these activity costs with the total project cost.

One of the key tools for interpreting the results is correlation analysis, which helps evaluate which input variables have the most significant influence on output variables.

Using the Tornado Chart

We can visualize this relationship using the tornado chart, which allows us to:

- Rank input variables based on their impact.

- Identify which factors most significantly affect total cost.

To do this, simply:

1.  Go to cell B8 in the Outputs sheet.

2.  Select the "Tornado Chart" option.

3.  Observe the ranking of the most relevant input variables.

Conclusion

- Defining output variables is essential for interpreting Monte Carlo simulation results.
- Input variables allow us to analyze which factors have the greatest impact on the results.
- Using tools like the tornado chart, we can gain valuable insights for decision-making.
- In future modules, we will explore advanced interpretation of these charts and their practical applications.

With this approach, we achieve a structured, clear, and decision-oriented analysis in any Monte Carlo simulation model.

1. Introduction

Before running a simulation, it is essential to define certain parameters that determine how the process will be displayed and executed. In the Easy Rest Quantum software, these parameters are configured within the options ribbon using the parameters button.

2. Parameter 1: Samples for Graph Visualization

This parameter does not influence the final results but affects how the data distribution is visualized.

• Example 1: If we set this parameter to 100, when displaying the results, we will see 100 contiguous bars, with some separation between them.

• Example 2: If we change the value to 800, the distribution will appear completely dense and continuous, allowing better visualization in tools like Excel.

3. Parameter 2: Number of Iterations

This is a crucial parameter as it determines how many times the simulation is executed.

• In the free student version, the maximum allowed number of iterations is 200.

• In professional versions (Easy Rest Quantum beginner and Luis Quantum), up to 50,000 iterations are allowed.

Practical Example: If we perform a stock price simulation with:

• 100 iterations: The distribution is limited and not very precise.

• 50,000 iterations: The simulation is more robust and representative.

4. Boolean Parameters

These parameters determine whether certain additional statistics will be displayed alongside the simulation histograms. They can be toggled on or off depending on user needs.

5. Percentile Positioning

There are two vertical delimiters (blue lines) that indicate different percentile levels:

• Percentile 1: Set at target value 1.

• Percentile 2: Set at target value 2.

The software automatically recalculates these values in the parameter template.

Practical Example: If we set a percentile of 1% and 99%, we can analyze extreme values in a financial simulation to understand the probability of obtaining atypical returns.

6. Number of Intervals for Histograms

Users can configure the number of intervals into which the histogram information will be divided:

• Minimum: 10 intervals.

• Maximum: 99 intervals.

This parameter only affects how the histogram accumulates, without modifying the original data.

7. Running the Simulation

Once the parameters are defined, the software will request confirmation to execute the simulation:

1. If accepted, the simulation will run immediately with the established parameters.

2. If declined, the user can modify the parameters before pressing the "Simulate" button and proceeding with the Monte Carlo simulation execution.

Lesson Summary

• Parameters affect visualization, iterations, statistics, and percentiles.

• The number of iterations varies depending on the software version.

• Histograms can be set between 10 and 99 intervals.

• The software prompts for confirmation before executing the simulation to ensure the parameters are correct.

With this understanding, users can optimize their simulations and obtain clearer and more precise visual results.

1. Introduction

A frequently asked question in Monte Carlo simulations is:

How many iterations are necessary to obtain precise and stable results?

The answer is not straightforward and depends on several factors, including the required precision, computational capacity, and the nature of the model.

In this lesson, we will explore how to determine the optimal number of iterations, balancing accuracy and computational efficiency.

2. Impact of the Number of Iterations on Processing Time

Relationship between iterations and execution time

• The higher the number of iterations, the longer the processing time.

• In an experiment testing 9 sets of iterations (ranging from 100 to 50,000), processing time increased linearly with the number of iterations.

Example:

• 100 iterations →  8 seconds

• 50,000 iterations → 2,600 seconds

Observed processing speed

• Between 13 and 24 iterations per second were generated, regardless of the dataset size.

• This indicates that simulation time grows proportionally with the number of iterations.

3. Factors Determining the Number of Iterations

The optimal number of iterations depends on various factors, such as:

- Model complexity (number of variables and relationships)

- Type of distributions used (symmetric vs. asymmetric)

- Presence of correlations between variables

- Level of precision required in results

4. Accuracy and Data Convergence

As the number of iterations increases:

- Precision improves → Reducing the variability of the results.

- Convergence stabilizes → Values stop changing significantly.

When Does Stability Occur?

Mean (average of data)

• After 5,000 iterations, the mean begins to stabilize, and changes very little with additional iterations.

Percentiles (extreme values of the distribution)

• In the 95th percentile, values still fluctuate with fewer than 1,000 iterations.

• Between 5,000 and 10,000 iterations are needed for the percentile to converge and stabilize.

5. Practical Recommendation: How Many Iterations to Use?

General rule

• For most simulations, a range between 5,000 and 10,000 iterations is sufficient to achieve convergence and stability without unnecessarily increasing processing time.

Histogram and graphical stability

• Observe, for example, this sequence of nine histograms generated with an increasing number of iterations, showing how the convergence or stability of the responses consolidated by the Monte Carlo simulation process improves. This series of graphs is generated with 100, 200, 500, 1000, 2000, 5000, 10000, 20000, and 50000 iterations.

• With 1,000 iterations, histograms still appear fragmented and poorly defined.

• With 10,000 iterations, histograms smooth out and converge, indicating better statistical representation.

6. Conclusion

- The higher the number of iterations, the higher the precision, but also the longer the simulation time.
- After 5,000 iterations, results begin to stabilize, with minor changes beyond 10,000 iterations.
- Percentiles require more iterations than the mean to converge.
- For most models, between 5,000 and 10,000 iterations is an optimal balance between precision and computational efficiency.

Final Recommendation: If computational capacity allows, using 10,000 iterations will ensure stable results without excessively increasing simulation time.

1. Introduction

Before executing a simulation, we can generate a series of statistics that will provide key information about the output variables. These statistics are simulation functions that allow us to better analyze the obtained results.

In the "Outputs" sheet, specifically in the range B2:B8, we have defined seven output variables that we want to monitor. Each of them can be associated with a set of statistics using functions from Easy Rest Quantum.

2. Calculating the Mean of an Output Variable

To obtain the mean of an output variable, such as the completion date, we use the function:

Where:

• $B$2 refers to the column where the completion date values are located.

• Once the simulation is complete, this function will update the value with the mean of the dates generated in different iterations.

Note: Initially, these values will display errors, as there are no simulated data yet to calculate statistics.

3. Calculating Percentiles

We can define a custom percentile in cell D2 using the function:

Where:

• B2 is the output variable.

• Paramex!B4 is the cell where we define the desired percentile (value between 0 and 100).

Once the simulation is complete, this value will represent the corresponding percentile within the set of obtained results.

4. Obtaining the Minimum and Maximum Value

To determine the minimum and maximum values recorded in the simulation, we can use the following functions:

• Minimum value (E2):

• Maximum value (F2):

Practical Example: If we are simulating a project's completion time, the minimum will represent the fastest recorded scenario, and the maximum will represent the latest case.

5. Calculating Probabilities with a Target Value

If we have a target completion date, such as July 25, 2026, we can calculate the probability of reaching it using:

Where:

• B2 is the output variable (simulated completion date).

• G2 is the target date (July 25, 2026).

Interpretation: Once the simulation is executed, this value will indicate the probability that the simulated date is less than or equal to the target date.

6. Extending Functions to Other Variables

To optimize the process, we can copy and extend this set of functions to the remaining six output variables. This way, all of them will update automatically once the simulation is completed.

7. Running the Simulation and Final Analysis

Initially, these functions will display errors because there are no simulated data yet. However, after running the simulation:

• The database will be populated with simulated values.

• The defined statistics will update.

• We will obtain a complete statistical profile of the behavior of the seven output variables in our project.

Conclusion With this procedure, we can analyze simulation results in greater depth and make informed decisions based on key statistics such as mean, percentiles, extreme values, and probability of achievement.

Introduction

In this lesson, we will learn how to execute our first probabilistic simulation in a project. We will explore the variability in task durations and costs, without including risks at this initial stage. This will lead us to the point where, finally, we will see how to interpret the results obtained, which is the ultimate purpose of everything we have developed so far.

1. Preparing the Simulation

Before starting, we must ensure that the scenario type we are generating is probabilistic. If we used a deterministic approach, there would be no variability in tasks, and the results would always be the same.

Follow these steps:

1. Verify parameters: Ensure that the scenario is probabilistic.

2. Observe uncertainty: Task durations and costs may have low, medium, or high uncertainty, which will affect calculations.

3. Check for changes: Every time we recalculate (e.g., by pressing F9), we will see different generated scenarios.

2. Effect of Uncertainty on the Project

Uncertainty impacts two key aspects:

Task Duration

• Depending on the level of uncertainty, durations will vary.

• These durations are found in column S of our spreadsheet.

• The project’s completion date, located in cell F1, will change with each simulation.

Fixed and Variable Costs

• Costs can also be uncertain.

• These values are found in the following columns:

  • G: Fixed costs.

  • N: Variable costs.

• These costs depend on task duration (column M).

So far, we have considered uncertainty in task duration and costs, but we have not yet included risks.

3. Temporary Exclusion of Risks

For this initial simulation, we will not include risks. Follow these steps:

1. Go to the Risk Sheet.

2. In column B, ensure that each risk has a value of zero.

3. This means that risks will not affect the project’s duration or costs in this simulation.

4. Simulation Setup and Execution

Before running the simulation:

• Set 5000 iterations. This is the minimum number needed to achieve stability and convergence in results.

• Start the simulation and wait a few minutes.

5. Analyzing the Results

Once the simulation is complete:

1. Review the "Outputs" Sheet:

   • Here, we will find a table with statistical values and compliance rates.

2. Explore the 7 Simulated Variables:

   • The simulation results are stored in the "Data" sheet.

   • This sheet contains 5000 rows, one per iteration.

   • We can access this data using the simulated data button.

Conclusion

We have successfully executed our first probabilistic simulation, considering uncertainty in task durations and costs. However, we have not yet incorporated risks. In the next phase, we will focus on interpreting this data and including risks in our analysis.

Introduction

We've reached the most interesting part of the course!
After running our simulation, we’re ready to interpret the results.
This analysis will be done step by step, starting with histograms, a key tool to visualize data variability.

1. What Will We Analyze First?

We’ll look at two of the 7 output variables found on the "Outputs" sheet:

• Total project duration in days (Cell B3)

• Total project cost (Cell B8)

2. Generating the Histogram

Project Duration

1. Position the cursor on cell B3

2. Click on the Histogram icon

3. A segmented bar chart will appear showing the 5,000 simulation iterations

Displayed information:

• Minimum: ~134 days

• Maximum: ~164 days

• Average: ~149 days

• Median (Percentile 50): ~149 days

• Default percentiles: 80% and 95%

To change the percentiles:

• Close the histogram

• Go to Parameters

• Change them to, for example, Percentile 75 and 90

• Regenerate the histogram

Now you’ll see two blue vertical bars at:

• 75th Percentile (~152 days): 75% chance the duration is ≤152 days

• 90th Percentile (~155 days): 90% chance the duration is ≤155 days

3. Customizing the Histogram

Want more detail?

• In Parameters, change the number of intervals to 99 (the maximum)

• Regenerate the histogram

• No need to rerun the simulation

The data is the same but more segmented, allowing for a finer visual analysis.

4. Analyzing Total Cost

Now repeat the process for cell B8 (Total cost):

• Minimum: ~20,700

• Maximum: ~27,500

• Average: ~23,801

• 75th Percentile: ~24,500

• 90th Percentile: ~25,200

This gives you a clear view of the cost variability in your project.

5. Why Do Results Change with F9?

Every time you press F9, new random values are generated, so:

• Cells from B2 to B8 change

• We don’t make decisions based solely on those cells

That’s why the analysis is based on:

• The histograms

• Or the statistics in columns C to H for each variable

6. Applying the Analysis

With the current data:

• If we go back to Parameters, we can insert an average completion date, for example:

  • July 28, 2026 in cell D5

• And an average cost of 23,804 in the appropriate field

These values reflect the effects of uncertainty (risks not yet included).

Conclusion

In this lesson, you learned how to:

• Use histograms to visualize results

• Interpret percentiles and averages

• Customize charts without rerunning simulations

• Understand why values change when recalculating (F9)

In the next lesson, we’ll learn how to fully interpret these results and what happens when we include risks in the analysis.

Introduction

In this lesson, you’ll learn how to use the simulator’s built-in statistical functions to quickly and accurately interpret simulation results. We’ll explore how to work with histograms, percentiles, and the powerful cumulative probability function (Meta).

1. Accessing Histograms

As we saw previously:

• Simply position the cursor over any cell defined as an input or output variable, such as those from B2 to B8.

• Once selected, you can generate the histogram, as long as the simulation has already been run.

2. Statistical Functions in the Histogram

Once the histogram is generated, you’ll also see a statistics table displaying values such as:

• Minimum

• Maximum

• Average

• Defined percentiles

But in addition, these same values can be automatically calculated in other cells using specific functions.

Example:

For the total cost in row 8, the function ISYMIN calculates the average value, which in this case is 23,804.
You’ll notice that this matches the value shown in the histogram exactly.

You can also use:

• MIN to get the minimum value

• MAX to get the maximum value

3. Using Custom Percentiles

By default, percentiles are defined in column D, such as the 80th Percentile.
If you want to use a different percentile:

1. Go to the Parameters sheet

2. Change the desired percentile in cell B4

3. Return to the Outputs sheet

4. Now you’ll see, for example, the 75th Percentile, which might correspond to 24,500

4. Meta Function: Inverse Cumulative Probability

A very powerful tool is found in column H, especially in cell H1.

What does this function do?

• It lets you define a target value on the X-axis of a variable (e.g., cost).

• The simulator will calculate what cumulative probability there is that results will be less than or equal to that value.

Practical example:

• If your deterministic scenario estimates total cost at 23,000 (see cell O1 in the "Costs" sheet)

• Enter that value into the Meta cell

• The simulator will use the ISI.TARGET function to calculate the probability that the simulated cost is ≤ 23,000

Try changing the value:

• If you enter 30,000, the probability will be 100%

• If you enter 20,000, the probability will be 0%

5. Conclusion

The Target column works interactively with the Probability column and provides:

- Instant percentile readings
- Comparison with reference values (like the deterministic scenario)
- Fast interpretation without manually analyzing charts

These tools allow for much more informed and precise decision-making once the simulation has been completed.

Introduction

In this lesson, you’ll learn how to interpret the duration-cost scatter plot, also known as the joint confidence level chart. This plot helps you visualize, in a single diagram, the relationship between two of the most critical variables in any project: the project’s completion date and its total cost.

1. What Does the Scatter Plot Show?

The plot compares two variables:

• X-axis (horizontal): Project completion date or duration

• Y-axis (vertical): Project total cost

This chart allows us to analyze how both variables behave simultaneously, which is essential for effective risk management.

2. The 4 Possible Scenarios

When analyzing the scatter plot, the simulation results fall into four possible quadrants:

Scenario

Time

Budget

Total success

On time

Under budget

Cost overrun

On time

Over budget

Delay controlled

Late

Under budget

Worst case

Late

Over budget

3. Visualization in Sheet “Scatter”

The "Skater" sheet displays this chart with the data points generated by each simulation (e.g., 5,000 iterations). Each dot represents a possible project scenario based on completion date and cost.

These data points are compared against two reference values found in cells B5 and B6 in the "Parameters" sheet:

• B5 → Target completion date

• B6 → Target budget

By default, these values correspond to a deterministic scenario, for example:

• July 25, 2026 as the expected finish date

• $23,000 as the baseline budget

4. Interpreting the Results

Each of the 5,000 data points will fall into one of the four quadrants.

Example of simulated scenario distribution:

• On time and under budget → 14%

• On time and over budget → 10%

• Late and under budget → 9%

• Late and over budget → 68%

This tells us that, in this case, more than two-thirds of the time, the project ends up late and over budget.

5. What Does This Mean?

This shows that the deterministic scenario (July 25 and $23,000) is too optimistic. In real life, it is unlikely to happen due to the uncertainty in task durations and costs.

Conclusion: We need more realistic targets.

6. Using Percentiles to Set New Goals

A common strategy is to use the 80th percentile to set targets with a higher chance of success.

Suggested new targets:

• Completion date: July 31, 2026 (80th percentile)

• Budget: $24,680 (80th percentile)

These values are entered in cells B5 and B6 in the “Parameters” sheet.

7. Change in Success Probability

After applying the new goals and updating the chart:

The probability of success (meeting both targets: time and cost) increases to:

68% probability of success

This means that if we use more realistic percentiles, 2 out of 3 times the project will succeed.

8. Why Isn't It Exactly 80%?

Even though both values (time and cost) are set at the 80th percentile, the combined probability drops to 68% because:

• It’s harder to meet two conditions simultaneously

• The combined probability reflects the conditional intersection of two events

Final Conclusion

Using the scatter plot and adjusting goals with percentiles allows us to:

• Visually understand project risks

• Set more realistic expectations

• Make better-informed decisions

We haven’t yet incorporated external risks into this simulation. But even so, this shows how probabilistic analysis dramatically changes the probability of project success.

In this part of the course, we’ll learn how to use analytical tools like the tornado diagram and the sensitivity map, which help us visualize the relationship between input variables (such as risks and task costs) and the output variables of the model, like the project’s total cost.

This visualization allows the project manager to establish the priority or hierarchy of risks and tasks, enabling a better allocation of resources to the most critical elements for the project’s success.

Input and Output Variables

We already have the output variables defined, located in the model’s “Outputs” sheet, specifically in cells B2 to B8. These represent the key results we will evaluate.

However, we haven’t yet defined our input variables—those that directly influence the model’s results.

To do this, we remove any previous configuration using the “Input Data” button, under the “Remove Input” tab, leaving the selection blank. By doing so, the iziRisk Quantum software will delete all previously defined input variables. We can confirm this step by clicking the “Show Inputs and Outputs” button, which should now display an empty space.

Defining New Input Variables

From the “Map” tab, we begin setting up the input variables. The first ten rows correspond to the project’s 10 main risks. In cell C3, we reference the stochastic behavior of a specific risk located in the “Risks” sheet, column T.

This formula is then copied down to row 12 to cover all 10 risks.

It’s important to note that these cells will change value every time we recalculate the simulation, as they are stochastic.

Incorporating Task Costs

We now turn to the costs associated with tasks. In the “Costs” sheet, you’ll find the fixed and variable costs of the project’s 28 tasks. This variability includes uncertainty in both costs and task durations.

In column O, starting at cell O3, we find the total cost per task, which includes both fixed and variable components. This is copied down for 27 rows, covering all 28 tasks.

We now have 10 risk-related variables and 28 task-related variables, all ready for simulation.

Calculating Total Cost and Means

In cell C2, we sum the values to obtain the total project cost per iteration. This value will change every time a new simulation is run.

Next, in column D, we use the function =isim(C2) to calculate the mean (expected) value of each variable. This is copied down for all rows of risks and tasks.

It’s normal for these cells to initially return errors if no simulation has been run yet or if the cells haven’t been formally declared as input or output variables.

Calculating Relative Contribution

In cell E3, we define each variable’s percentage contribution by dividing its mean value by the total project cost. This formula is copied down, resulting in a relative impact measure for each variable.

Ranking Importance

We now apply the function =RANK(D3, $D$3:$D$40) to generate a numerical ranking, from 1 to 38, sorting all variables by their contribution to the total cost. This tells us which risks and tasks have the greatest influence on project results.

Final Declaration of Input Variables

We select the range of cells from D3 to D40, which contain all relevant variables. Then we go to the “Input Data” button and confirm the selection.

Thanks to the labels in the adjacent column, iziRisk Quantum will automatically assign the correct names to each variable.

At this point, the model is fully ready to run simulations.

Simulation and Analysis

We run a simulation of 1,000 iterations using the “Parameters” button or directly through “Simulate”. This will generate and store the results for all declared input variables.

In this case, we’re working with mean values, as summing percentiles is not mathematically correct.

Once simulations are complete, we’ll have:

• In column D: the mean values of costs

• In column E: their relative contributions

This information is now ready to be visualized graphically.

Visualizing Results

The information is displayed in a map-style chart, where we use conditional formatting to highlight the most relevant elements. For example, we see that tasks 23, 24, and 25 together account for 54% of the total project cost.

Other highly impactful elements include:

• Technical and integration issues

• Database design

This chart gives us a visual and quantitative view of the most significant risks and tasks in terms of their economic impact.

In simple terms, correlation is a measure that tells us how closely one variable is related to another.

In the context of project management, it helps us answer questions like:
How much does a specific task with its uncertainty or a risk influence the total project cost?

What does a correlation coefficient mean?

• Correlation is a number between –1 and +1.

• When it’s positive and close to 1, it means that when that task becomes more expensive, the overall project also tends to become more expensive.

• When it’s close to 0, the task has little to no relationship with the total cost.

• If it were negative (which is not the case here), it would imply that when that task becomes more expensive, the project cost tends to decrease — rare, but possible under specific conditions.

What is a correlation tornado?

The correlation tornado is a chart that orders all project variables (tasks or risks) from highest to lowest based on how strongly they are related to the total cost.

It’s called a “tornado” because of its funnel-like shape:
the longest bars (highest correlations) appear at the top, and smaller bars descend below.

This chart allows you to quickly see:

• Which tasks or risks have the greatest impact on the total cost

• Where you should focus attention or apply stricter controls

The most influential variables in your project

In the Outputs sheet, place your cursor over cell B8, which contains the output variable “Total Cost.”
Remember that this cell was previously declared as an output variable.

Also, since the simulation has already been executed, the model contains all the iterations stored in the hidden sheet Data, which you can access (or hide again) using the Simulated Data button.

• Column B in Data contains all the iterations of the Total Cost variable.

• To the right, you’ll find stored iterations for the input variables: task costs and risks.

This database is what allows the calculation of correlation coefficients.

Once your cursor is placed over B8, click the Correlation Tornado button.
If the selected cell is not an input or output variable, a tornado chart cannot be generated.

In this analysis, the correlation between each individual cost and the total project cost was calculated.

The five variables with the highest positive correlation were:

1. Development Tasks (correlation: 0.54)
   → When these tasks get more expensive, the total project cost increases significantly. They are the main impact factor.

2. Backend Development (0.52)
   → Very strong correlation. Any deviation at this stage directly affects the overall budget.

3. Frontend Development (0.27)
   → Still influential, though to a lesser degree. Still worth monitoring.

4. Risk of Cost Overruns and Poor Budget Management (0.27)
   → This risk shares the same correlation level as frontend development, making it a key focus area.

5. Risk from External Dependencies and Vendor Delays (0.25)
   → This factor also has a clear influence. Delays in supplies or third-party services can seriously compromise final project costs.

What can you do with this information?

• Prioritize monitoring and planning for these tasks and risks.

• Apply more frequent quality controls and tracking in Backend Development and budget management.

• Consider mitigation actions for external dependencies, such as contracts with delay penalties.

Introduction to Scatter Plots

A great way to visualize the relationship between input variables (such as tasks and risks) and an output variable (such as total project cost) is through a scatter plot. This type of chart helps us understand how different variables affect the final project outcome.

Starting Point: The Output Variable

We start with the most economically important output variable: total cost. This is located in cell B8 of the 'Outputs' sheet. From there, we can click the 'Correlation Tornado' button to see all variables ranked by their correlation coefficient with total cost.

Example: Task with High Correlation

In the example, the development task has the highest correlation coefficient. By selecting this variable and clicking the 'Scatter Plot' button, we see a cloud of points with a blue line that has a positive slope. This indicates that when the cost of this task is low, the total project cost also tends to be low, and vice versa. The correlation coefficient in this case is 0.54, showing a moderately strong relationship between the two variables.

Example: Task with Low Correlation

Now we take a task with a very low correlation coefficient, such as the design task. When plotted against total cost, the blue line is nearly flat. This means that regardless of whether the task cost is high or low, it does not significantly impact the total project cost.

Common Misinterpretations

A common mistake is to believe that correlation coefficients can be added up to get a total or percentage distribution. That is not correct. Each coefficient must be interpreted individually. They do not represent proportions and cannot be summed to 100%.

Conclusion

Scatter plots clearly show which variables truly impact project outcomes. Along with the correlation tornado, they are key tools to prioritize tasks and risks based on real data, not assumptions.

What is an S Curve Chart?

An useful way to visualize variables with uncertainty or variability —such as tasks, risks, or costs— is through the S Curve chart, also known as a cumulative or percentile chart.
Do not confuse it with the S Curve used in project management to show progress over time. Here, the S Curve shows the probabilistic distribution of outcomes.

How to Access the Chart in iziRisk Quantum

1. Go to the 'Outputs' sheet.
2. Locate cell B8, which contains the project's Total Cost.
3. Click the *Histogram* button to view the distribution.
4. You’ll see the histogram curve and a table to the right with stats like minimum, maximum, mean, and percentiles.

Differences Between Histogram and S Curve

• The histogram’s vertical axis has no units, as probability is measured by the area under the curve.
• The S Curve chart instead shows cumulative probability.
This allows you to see how possible total cost values are distributed and accumulated.

Multi-Curve Visualization with S Charts

One great advantage of the S Curve chart is that you can overlay multiple curves to visually compare different components:
• Total fixed costs of activities
• Variable costs
• Total risk costs
Clicking the S Curve button and selecting these variables lets you see all the curves in one chart.

Interpreting Overlaid Curves

• The *risk cost* curve (green line) appears on the left: lower magnitude and greater variability.
• The *fixed* and *variable cost* curves appear further to the right with less spread.
• A fourth curve —the *Total Cost*— appears even further right, summarizing the others.

Conclusion

The S Curve chart offers a richer interpretation of model uncertainty.
It’s ideal for comparing, analyzing, and prioritizing project components with uncertain economic impact.

1. Why Is It Important to Estimate a Proper Project Contingency?

Contingency estimation is a key part of professional project planning. An adequate contingency allows project managers and decision-makers to:

• Negotiate confidently with stakeholders, showing realistic and justifiable cost or time reserves.

• Protect the project’s success, ensuring that even with uncertainties, it can be delivered on time and within budget.

• Align expectations between project sponsors, clients, and technical teams.

• Avoid financial overruns or reputational damage caused by unrealistic underestimation of uncertainty.

• Build trust, transparency, and accountability through data-based forecasts.

2. Different Methods for Estimating Contingency

There are several ways to estimate contingency. Below is a comparison of the main approaches:

In this document, we focus on the probabilistic method using Monte Carlo simulation, as it is the most data-driven, transparent, and robust approach.

3. What Is Probabilistic Contingency (According to PMI)?

The Project Management Institute (PMI) defines probabilistic contingency as a reserve calculated using quantitative risk analysis, often through Monte Carlo simulation.

From the PMBOK® Guide (7th Edition):

"Contingency reserves are estimated using quantitative methods such as Monte Carlo simulation to address identified risks."

How is it calculated?

• A simulation model is built with all the cost or schedule elements.

• Uncertainties are modeled using probability distributions.

• Thousands of scenarios are simulated.

• A percentile is selected (e.g., P70), and the contingency is the difference between that percentile and the expected value. The expected value is the mean.

4. Why Use Probabilistic Contingency?

Using a probabilistic contingency allows:

• Confidence-based decision making.

• Transparency for stakeholders.

• More accurate reserve levels tailored to project uncertainty.

• Support for risk-informed governance.

Unlike fixed percentages or expert guesses, probabilistic methods justify the reserve based on simulated outcomes.

5. Typical Percentiles by Industry

These references are based on industry guidance by PMI, AACEi, NASA, DOE, IPA, and UK government publications. Full references can be found in the tab “Contingency.”

6. Step-by-Step: How to Calculate Probabilistic Contingency

To calculate contingency in iziRisk Quantum, follow these steps:

a. Hover your cursor over the cell Outputs!B8 which contains the Total Cost output.

b. Click the Histogram icon to generate the probability distribution.

c. Check column D, where percentiles are shown. D8 gives the selected percentile (e.g., P70).

d. Choose the appropriate percentile from the industry table above. For example, for IT projects, use 70th percentile located in Parameters!B4.

e. Read the 70th percentile value in Outputs!D8.

f. In Outputs!D9, calculate Contingency = D8 – C8 (i.e., Percentile minus Mean).

g. In C9, calculate the relative contingency as =D9/C8.

h. Final result: “The probabilistic contingency at the 70th percentile is $X and represents Y% of the project’s total cost.

i. This same procedure can be done for project duration, using the cell B2 that contains Completion Date.

7. Conclusion

Probabilistic contingency estimation using Monte Carlo simulation provides:

• Better insight into how risks affect project outcomes.

• Quantitative justification for reserves.

• A confidence-based approach aligned with industry best practices.

By choosing an appropriate percentile and following a structured method, project teams can enhance planning quality, manage expectations, and improve the chances of on-time and on-budget delivery.

In the second section of the Monte Carlo Simulation Introduction course, we compared the PERT and Triangular distributions. In lessons 12, 13, and 14, we analyzed their differences but left a definitive conclusion pending. We will now complete that comparison.

DISTRIBUTION SETUP

To do this, we go to the Parameters sheet, cell B8, and first select the PERT distribution.

This distribution, like the Triangular one, requires three parameters: a minimum, a mode (most likely value), and a maximum. The difference lies in the shape of the curve: PERT is smoother, while Triangular has straight lines.

When selecting PERT, it is applied to:

• Task durations

• Fixed and variable costs

• Risk impacts (in both duration and cost)

SIMULATION WITH PERT

We run a simulation of 20,000 iterations and store the results in the Outputs sheet, specifically in column I, using the 70th percentile as a reference—since this is the recommended level for IT projects based on common practices discussed in the previous lesson.

SIMULATION WITH TRIANGULAR

Then, in cell B8, we change the distribution to Triangular and run the simulation again with 20,000 iterations. We save these new results in column J to compare them with the previous ones.

RESULTS COMPARISON

Note: Remember that based on the randomness principle of Monte Carlo simulation, the results reported here will likely differ slightly from those simulated at different times or by other users—especially if the number of iterations is low.

Now we have:

• Column I: Results with PERT

• Column J: Results with Triangular

• Column K: Percentage comparison between both

For example:

• With PERT, the project finishes on August 8, 2026

• With Triangular, it finishes on August 13, 2026

That’s a 5-day difference, which might seem minor but affects total duration:

• PERT: 161 days

• Triangular: 165 days

The same applies to costs:

• PERT: $30,923

• Triangular: $32,139

This represents an increase of nearly 4% in costs and 2.5% in duration.

WHICH DISTRIBUTION TO USE?

It will be up to the analyst’s judgment to decide whether these differences are significant enough to prefer one distribution over the other. In lessons 12 to 14, we discussed the pros and cons of both.

IMPORTANCE OF THE NUMBER OF ITERATIONS

This analysis was conducted using 20,000 iterations in both cases. But what happens if we only use 200?

Simulations with a low number of iterations (e.g., 200) result in fragmented, less precise graphs. This often occurs in student or free versions of the software that have limitations.

When we increase the number to 20,000 iterations, the results become more stable. To test this, we ran 10 simulations with only 200 iterations and compared:

• The mean result was 31,061

• The 70th percentile was 32,186

Based on this data, we calculated the coefficient of variation:

• For the mean: only 0.46%

• For the 70th percentile: 0.72%, which in at least one case represented an overestimation of contingency by about 2% when using only 200 iterations

This shows that while the mean stabilizes quickly, percentiles require more iterations for better accuracy.

GENERAL RECOMMENDATION

Attached charts show histograms based on 200 and 20,000 iterations respectively.

Our recommendation is:

• For educational purposes, a rough approximation using just 200 iterations is acceptable.

• For professional use, we recommend using at least 5,000 to 10,000 iterations as a practical rule of thumb to ensure reliable convergence in project or business simulation models.

In this part of the course, we are going to integrate everything we have worked on so far with the model, starting from the beginning.

1. DETERMINISTIC SCENARIO

We began with a deterministic scenario, meaning a model without variability. In this case:

• Tasks do not have uncertainty in duration or cost.

• No risks are considered.

• Everything behaves predictably, just as planned.

Results from this scenario:

• Finish date: July 25, 2026

• Total duration: 146 days

• Total project cost: $23,080

  • Fixed cost: $12,080

  • Variable cost: $11,000

These values can be recorded in cells C6 and C7 of the "Parameters" sheet.

In this scenario, by definition, no probabilities can be calculated. There is no way to know the likelihood of meeting time and cost objectives.

2. SCENARIO WITH UNCERTAINTIES

Next, we took a step forward and incorporated uncertainties:

We did this by changing the Scenario Type Parameter in cell Paramets!B7 from Deterministic to Probabilistic, generating a new simulation, and then observing the results in column D of the Outputs sheet using the 70th percentile as our reference.

• These reflect variability in task durations and costs.

• No specific risks are identified, but it is assumed that random factors may affect project execution.

Configuration:

• We changed the scenario type from Deterministic to Probabilistic.

• We used the PERT distribution to represent variability.

• We ran a simulation of 5,000 iterations.

• We used the 70th percentile, the typical contingency level for IT projects.

Results:

• New finish date: July 30, 2026 (5 days more than the deterministic case).

• New total cost: $24,801

  • This represents an increase of $1,700, or 7.5% more than the scenario without uncertainties.

We now record these results in cells D5 and D6 of the Paramets sheet, under the scenario “Probabilistic with Uncertainties Only.”

In contrast, the original deterministic scenario would naïvely assume:

• 0% probability of finishing on time and within budget

• 5% probability of being on time but over budget

• 1% probability of being late but within budget

• 94% probability of missing both time and budget targets

3. SCENARIO WITH UNCERTAINTY + RISKS

Finally, we incorporated identified risks into the model:

• In the “Risks” sheet, we activated the 10 included risks by marking them with a 1 in the corresponding column. We then re-ran the simulation and observed the results in column D of Outputs at the 70th percentile.

We considered two types of risks:

• Activity risks: affect specific tasks (duration and cost)

• Cross-cutting risks: affect the project globally, usually only in terms of cost

Simulation:

• We ran 5,000 iterations again with these risks included

Results:

• Finish date: August 21, 2026 (27 days later than the initial scenario)

• Total cost: $31,100

  • This represents a 35% cost overrun compared to the deterministic model

4. WHAT NOW?

We have now seen how the model evolves when we include:

1. Uncertainties

2. Risks

3. Simulations with different distributions and scenarios

If we had committed to the deterministic date of July 25, 2026 and the budget of $23,080, we would have faced the following probabilities:

• 0% probability of finishing on time and within budget

• Only 1% probability of being on time but over budget

• 0% probability of being late but within budget

• And a 99% probability of missing both time and budget

In previous sections, we also identified — thanks to the correlation tornado — the most relevant project risk:
Technical and integration problems

5. NEXT SECTION: RISK MITIGATION

We now ask the following question:

What would happen if that main risk didn’t exist?

We will conduct a marginal analysis to study:

• How results would change if we eliminate that risk

• What mitigation strategies could be applied

• How to reduce its negative impacts

This leads us to the final part of the course:
Formulating action plans based on mitigation strategies.

In this lesson, we will learn how to identify, model, and assess a specific risk: technical and integration problems in a project. This is the most significant risk in the project. We will do this using two alternative approaches to estimate the cost of the risk: direct analysis and incremental analysis. This will lead us to strategies to mitigate it.

1. RISK IDENTIFICATION

As we saw in Lessons 7.4, we can detect the most significant risks using a correlation tornado chart on cell B8 (total cost). In this analysis, the second most impactful factor is the risk of technical and integration problems, only surpassed by the development task.

Another way to identify it is through the risk map sheet, discussed in Lesson 7.5, where this is Risk number 7, and it is among the four risks with the highest potential impact according to the 70th percentile.

2. RISK ASSOCIATION WITH TASKS

Let’s explore the usefulness of the RiskEx sheet to visualize risks. Select a 7 in cell A2 of RiskEx. This risk is linked to Task 16. To assess it, we model two key elements:

• Frequency: the number of times it may occur during the project. It is modeled with a Poisson distribution with a mean of 2.

• Impact on duration: each event may delay the task between 2 days (minimum), 5 days (most likely), and 10 days (maximum), using a PERT distribution.

• Impact on cost: each event could cost between $500, $800, and $1,000, also modeled using a PERT distribution.

By combining frequency and impact (convolution), we obtain the total effect of the risk on the task. Remember that we saw the concept of convolution in lesson 2.17.

3. ESTIMATING THE COST OF THE RISK

OPTION 1: DIRECT ESTIMATION

• In the Risks sheet, we define cell T9 as an output variable, which contains the total cost of the "Technical and Integration Problems" risk.

• Then we simulate the model and analyze the histogram.

• The result shows that the average cost of the risk is $1,560.

• This means that any mitigation strategy costing more than $1,560 would not be economically efficient.

• At the 70th percentile, the cost could reach $2,110, which could also serve as a decision threshold.

OPTION 2: MARGINAL ESTIMATION

• First, we simulate the model with all risks activated and obtain the total cost, with a mean of $29,900.

• Then we exclude Risk 7, re-simulate, and record the new mean cost of $28,300.

• The difference between both results ($1,600) is the marginal value of the risk.

COMPARISON OF METHODS

Both methods may produce similar results, but marginal estimation is more robust when dependencies exist between tasks or risks. For example, if there are risk correlations, or successor tasks that—depending on a probabilistic schedule—may be delayed, etc. That is why we prefer the marginal method in complex models where correlations influence the results.

5. CONCLUSIONS

• Risk analysis can be done individually or marginally.

• The marginal approach is more comprehensive when interdependencies exist.

• Knowing the expected cost of a risk allows us to evaluate whether it’s worth mitigating and how much to invest.

• In the next lesson, we will apply this analysis to design a specific mitigation strategy for this high-priority risk.

In this lesson, we will learn how to evaluate a risk mitigation strategy using key quantitative criteria: frequency, impact, probability of success, and cost. This approach aligns with PMI guidelines and can be implemented using analysis tools such as simulation models.

1. PMI DEFINITION OF RISK MITIGATION

Risk mitigation involves taking action to reduce the probability and/or impact of a risk to an acceptable threshold. It is a proactive response strategy designed to lessen the effect of risks that could negatively affect project objectives.

According to the PMI (Project Management Institute) and the PMBOK® Guide (Project Management Body of Knowledge), risk mitigation is one of the response strategies for negative project risks (also known as threats).

2. EXAMPLES OF RISK MITIGATION ACTIONS IN PROJECTS

• Adding resources to critical path activities to reduce the chance of delays.

• Implementing more robust quality assurance processes to reduce the likelihood of defects.

• Designing system redundancies to minimize the impact of failure.

• Conducting training to reduce human error.

• Improving communication channels to prevent misunderstandings.

3. KEY POINTS

• Mitigation does not eliminate the risk, but reduces its potential harm.

• It is applied before the risk occurs—unlike contingency plans, which are reactive.

• It is part of the Plan Risk Responses process in PMI’s risk management framework.

4. KEY COMPONENTS FOR EVALUATING A MITIGATION STRATEGY

a) Impact Reduction

A strategy may:

• Reduce the duration of a task affected by the risk.

• Lower the associated costs if the risk event occurs.

b) Frequency Reduction

The strategy may reduce how often the risk event could occur during the project.

5. PROBABILITY OF SUCCESS

Not all strategies are 100% effective. It is important to assign them an estimated probability of success (e.g., between 0% and 100%).

Example:
Hiring a public relations firm to counter reputational damage. This strategy might have a 60% probability of success, which should be factored into the analysis.

6. IMPLEMENTATION COST

Mitigation strategies are not free. Their implementation involves costs, whether explicit (direct expenses) or implicit (use of internal resources).

The cost can be modeled as a probability distribution with:

• Minimum value

• Most likely value

• Maximum value

This allows uncertainty in mitigation costs to also be accounted for.

7. ANALYSIS SUMMARY

A mitigation strategy may:

- Reduce the frequency of the risk
- Reduce its impact (in time or cost)
- Have a probability of success less than 100%
- But it involves an additional cost

The ultimate goal is to determine whether the mitigation benefits outweigh the cost of implementation.

8. CONCLUSION

A good mitigation strategy is one that:

• Significantly reduces exposure to the risk

• Has a high probability of success

• Is cost-effective

In the next lesson, we’ll see how to incorporate these strategies into our simulation model and how to determine whether they are worth implementing.

In this lesson, we will learn how to evaluate the impact of a mitigation strategy on a specific risk within our simulation model. We’ll use an incremental approach, comparing three scenarios:

1. Without mitigation

2. Without the risk

3. With active mitigation

1. Starting Point: Baseline Scenario

We begin with the probabilistic scenario that includes uncertainties and risks, without applying any mitigation strategy.

• Ensure that all risks are activated.

• Run 5,000 simulation iterations.

• Analyze results at the 70th percentile.

Results without mitigation:

• Project finish date: August 20, 2026

• Total cost: $31,000

This serves as our reference scenario or baseline.

2. Scenario Without the Main Risk

We had previously identified Risk No. 7: Technical and Integration Problems as the most relevant.

We deactivate this risk in the risk sheet and run the simulation again.

Results excluding the risk:

• New finish date: August 9, 2026

• New total cost: $29,000

• Net savings: 11 days and $2,000

This is a hypothetical scenario where the main risk does not exist.

3. Applying a Mitigation Strategy

We now simulate the active mitigation of Risk No. 7.

In the risk sheet, the segment from columns U to AD allows us to control how we implement each component of a mitigation strategy.

• In cell U9, we enter a value of 1 to activate the mitigation strategy for this risk.

• We activate duration mitigation (V9) and cost mitigation (W9) with values of 1.

• We estimate the following effects:

Expected effects of mitigation:

• Frequency reduction: enter 75% in cell X9

• Duration reduction: enter 75% in cell Y9

• Cost impact reduction: enter 75% in cell Z9

• Probability of success: 90%, entered in cell AA9

A Bernoulli (binary) function simulates this probability: in 90% of iterations, the mitigation is successful; in the remaining 10%, the strategy incurs a cost but has no effect on duration or cost.

• Estimated implementation cost modeled as a distribution:

  • Minimum: $200 (cell AB9)

  • Most likely: $300 (cell AC9)

  • Maximum: $500 (cell AD9)

4. Results from the Mitigated Model

We run the simulation again with the active mitigation plan.

Results at the 70th percentile:

• Finish date: August 9, 2026

• Total cost: $29,567

• Net improvement in cost due to mitigation: $1,440

• Net time savings due to mitigation: 11 days

5. Probabilistic Performance Assessment

Simulating this mitigated scenario yields the following probabilities:

Scenario

Probability

On time and under budget  55%

On time but over budget  13%

Late but under budget  15%

Late and over budget  17%

Lesson Conclusion

This exercise shows that:

• Mitigating a risk can reduce both time and cost.

• The incremental analysis helps us understand the net effect of the risk.

• The probabilistic model helps evaluate whether it's worth investing in the mitigation strategy.

In the next lesson, we’ll explore how to compare multiple mitigation alternatives to choose the most efficient one.

INTRODUCTION

A key part of quantitative risk management in projects is the ability to compare different mitigation strategies and analyze their impact in terms of both time and cost. In this lesson, you will learn how to evaluate whether a more expensive mitigation strategy truly adds value to the project.

SCENARIO COMPARATIVE ANALYSIS

Let’s suppose we decide to apply an alternative mitigation strategy to address a relevant project risk. To do this, the following changes were made:

• The risk parameters in the simulation model were modified:

  • Its frequency was reduced by 95%.

  • Its estimated impact (both in cost and duration) was also reduced by 95%, reflecting the expected effect of the mitigation.

  • The project’s success probability was increased from 90% to 95%.

  • The direct mitigation cost was increased in the cost sheet to a fixed value of $500, for the minimum, most likely, and maximum values.

After making these adjustments, the Monte Carlo simulation was run again to observe how results change under this new mitigated approach.

SIMULATED RESULTS WITH MITIGATION

After re-running the model with these modified parameters:

• The project completion date moves forward by one day, now ending on August 8.

• The total project cost under this strategy is $29,084.

This outcome is practically the same as the base cost of the first mitigation strategy, which had a most likely cost of $300 instead of the $500 required by this second strategy. In other words, the maximum amount we’d be willing to pay for a higher-quality mitigation strategy—one that guarantees greater reductions in frequency and impact, and increases the probability of success—is this $500 ceiling. Any cost above this threshold would no longer be worth paying.

A TOOL FOR DECISION-MAKING

This type of analysis allows simulation to be used as a quantitative decision-support tool, enabling us to:

• Compare different mitigation alternatives.

• Evaluate their impact on both project duration and total cost.

• Determine which option provides the best economic return.

CONCLUSION

Simulations not only help us anticipate risks—they also allow us to economically quantify our mitigation strategies.

This approach enables objective decision-making, prioritizing those measures that truly add value in terms of time, cost, and overall project reliability.

OBJECTIVE OF THIS LESSON

In this lesson, you will learn how to evaluate the simultaneous fulfillment of two key conditions in project management: finishing on time and within budget.
We will use simulation techniques and reverse analysis in Excel to find optimal combinations of schedule and cost that meet a desired confidence level (e.g., the 70th percentile). Therefore, we will present the final results to the interested parties of the project.

STEP-BY-STEP: SIMULTANEITY ANALYSIS

1. Enter the target values:
   Input reference values into the appropriate model cells, for example:

   • Target date in cell B5

   • Target budget in cell B6

2. Evaluate the combined result:
   Even if you achieve a 70% probability of meeting the schedule or the cost separately, this does not guarantee a 70% simultaneous success rate.
   That’s because both conditions must be met together, and their joint probability is lower.

3. Solution: Set a date and use reverse analysis:
   Fix a desired project completion date in cell B5, for example, August 10.
   Then use Excel’s Goal Seek ("What-if Analysis") function to determine what budget value in B6 will make B16 (the joint success probability) reach exactly 70%.
   This will yield a cost value of approximately $30,900.

MODEL RESULTS

Upon executing this technique:

• It is determined that with a budget of approximately $30,900, the project can be completed by August 10 with a 70% probability of joint success (on time and within budget).

EXPLORE SCENARIOS THROUGH SIMULATION

Re-run the simulation using the adjusted conditions:

• Iterations: 10,000

• Results:

  • On time and within budget: 70%

  • On time but over budget: 3%

  • Late but within budget: 21%

  • Late and over budget: 6%

These results confirm that the selected configuration meets the desired probabilistic contingency level.

INTRODUCTION TO INDIFFERENCE CURVES

This analysis leads to the construction of an indifference curve: a set of combinations of finish date and budget that jointly satisfy a given confidence level (e.g., the 70th percentile).

• You can build a table of results and graph the combinations that meet the desired level.

• This is done by selecting multiple combinations and recording whether the model yields the expected joint success.

CALCULATING PROBABILISTIC CONTINGENCY

After running the simulation:

1. Calculate the contingency on the Total Cost output variable (cell Outputs!B8) as the difference between:

   • The value at the 70th percentile.

   • The mean expected value.

2. Express this difference as a percentage of the mean.

Example:

• 70th percentile value: $31,000

• Mean value: $30,000

• Contingency: $1,000

• Relative proportion: 3.6%

This is interpreted as a probabilistic contingency of 3.6% to ensure with 70% confidence that the project finishes on time and within budget.

FINAL REFLECTION

Thanks to this technique:

• We understand the value of modeling both key project goals simultaneously.

• We optimize planning decisions based on quantitative analysis.

• We visualize equivalent trade-offs using indifference curves.

1. Welcome to the Conclusion

Hello again, and congratulations. You’ve reached the final chapter of our course Project Risk Quantification with Monte Carlo Simulation. My name is Fernando Hernández, and it has been an honor to guide you through this comprehensive journey into the world of quantitative project risk management.

Over the past sessions, we’ve explored a vast array of concepts, tools, and practices that equip you to model, simulate, and analyze the risks that impact your projects.

2. A Summary of What You’ve Learned

Let’s take a moment to reflect on the key elements we’ve covered:

• The Three Pillars: You now understand how tasks, costs, and risks must be modeled and integrated—not in isolation, but in a unified framework that mirrors the true complexity of project management.

• Uncertainty and Risk Distributions: You’ve worked with empirical distributions like PERT and Triangular, as well as continuous ones like Log-normal and Exponential, to model the variability and impact of risks realistically.

• Monte Carlo Simulation: We’ve learned how this method allows us to run thousands of iterations—far beyond a simple “best-case vs worst-case” comparison—to generate meaningful distributions of outcomes for time, cost, and total risk exposure.

• Frequency-Severity Modeling: You’ve mastered how to decompose risks into how often they occur (frequency) and how big their impacts are (severity), and how to integrate them using convolution techniques.

• Risk-Based Decision Making: We discussed how to interpret percentiles, estimate contingencies, identify tail risks, and use simulation outputs to justify mitigation investments or schedule buffers.

3. Practical Tools You Now Have

Along the way, you’ve also acquired:

• Access to IziRisk Quantum, the Excel-based or cloud-based simulation tool.

• Templates for integrating risk with task durations and costs.

• Visualization tools like Gantt charts that incorporate uncertainty.

• Analytical techniques to calculate and justify contingency reserves using the P80, P90, or P95 logic.

4. A Final Word on the Mindset Shift

This course is more than a set of technical skills—it’s about a transformation in how we think about risk.

Too often, projects rely on subjective matrices and fixed estimates. But risk is dynamic, interconnected, and quantifiable. You've learned that with the right tools, risks can be measured, modeled, and managed.

You’ve shifted from intuition-based planning… to data-driven decision-making.

5. What Comes Next?

If you’ve completed the course exercises and reviewed the simulations, you are now prepared to:

• Apply Monte Carlo simulation in real project environments.

• Educate colleagues and stakeholders on the value of quantitative risk management.

• Continue exploring advanced topics, such as:

  • Risk correlation

  • Portfolio-level analysis

  • Probabilistic cost estimating in capital projects

  • Quantification of operational risks

  • Quantification of complex risks

6. Your Certificate Awaits

Participants who have completed all modules, reviewed the simulations, and submitted their final assessments are eligible to receive a Certificate of Completion. This certificate recognizes not only your time commitment but also your dedication to raising the standard of project risk analysis in your organization or profession.

7. Exclusive Software Access for Course Participants

As part of your learning journey in this course, you've had hands-on experience using two powerful tools developed by iziRisk:

• iziRisk Quantum: Our advanced Excel-based Monte Carlo simulation add-in.

• iziRisk Project: The intuitive, cloud-based platform for project risk quantification.

Take advantage of these special rates to elevate your risk management capabilities with the professional-grade tools you've already learned to use. Whether you prefer working in Excel or the cloud, we’ve got you covered.

8. Thank You and Farewell

Thank you for being part of this learning experience. I hope you walk away not just with knowledge, but with conviction—conviction that quantitative methods like Monte Carlo simulation are not just useful, but essential for managing modern projects effectively.

On behalf of the entire iziRisk team, I congratulate you and encourage you to keep exploring, questioning, and improving the way risks are managed in the projects that shape our world.

Until next time, this is Fernando Hernandez, wishing you clarity, confidence, and control in all your future projects.