# Monte Carlo simulation

Use a Monte Carlo simulation to account for risk in quantitative analysis and decision making.

A Monte Carlo simulation is a quantitative analysis that accounts for the risk and uncertainty of a system by including the variability in the inputs. The system may be a new product, manufacturing line, finance and business activities, and so on. The simulation uses a mathematical model of the system, which allows you to explore the behavior of the system faster, cheaper, and possibly even safer than if you experimented on the real system.

The simulation provides expected values based on equations that define the relationship between the inputs (X) and outputs (Y). These may be known equations, or they may be based on a model that you created from a designed experiment (DOE) or regression analysis in Minitab. Suppose you are investigating the time to complete similar construction projects. There are four phases: Proposal, Scoping, Execution and Delivery. The measurements are in business days. Here, the inputs are the number of business days it takes to complete each phase. The output is simply the total number of business days it takes to complete the project.

Project Time = Proposal + Scoping + Execution + Delivery

Here's how the simulation works:

1. To account for the variability in the number of days to complete each phase, you need to specify a distribution for each input to describe its variation. For the construction project, the inputs and distributions may be as follows:
• For the Proposal phase, any number of days between ½ a day and 2 days is equally likely, so you use a uniform distribution.
• For the Scoping and Execution phases, you can only estimate the minimum, maximum, and most common number of days. Therefore, you use the triangular distribution to describe these inputs.
• For the Delivery phase, the data follow a normal distribution, so you need to provide the average number of days (5) to complete the phase and the standard deviation (1.5).
2. Each iteration of the simulation draws a random sample of possible values from the distributions for each input, enters these values into the equation, and then calculates the outputs.
3. The results from all iterations are summarized to provide the expected values for the outputs.

Companion displays a histogram and summary statistics, including expected output values and an estimate of their variability. If you provide specification limits, the results also include process performance metrics.

Companion provides the following analysis methods to help you further improve the results of the initial simulation:
• Parameter optimization: Identifies optimal settings for the inputs that you can control. Companion searches a range of values for each input to find settings that meet the defined objective and lead to better performance of the system.
• Sensitivity analysis: Identifies the inputs whose variation have the most impact on your key outputs. Use this method along with your process knowledge to identify the inputs that can be adjusted to make improvements.
• What distribution best fits my input data? What values can I expect for my outputs?
• How capable is my process or product, given uncertainty in the input parameters?
• What are the optimal settings to achieve my goal?
• How does the variation in the inputs affect the variation in the response?
When to Use Purpose
Early-project Use simulation to predict how long a project will take.
Mid-project Given y=f(x), use simulation to estimate values for a response of interest.

## Data

This form has no data requirements because Companion simulates the data based on the model assumptions. However, if you're unsure which distribution to choose when defining the model inputs, Companion can evaluate historical data that is stored in a CSV file and recommend a distribution.

## Guidelines

A Monte Carlo simulation relies on the assumptions you make. First, you must obtain the equation (y= f(x)) that explains the relationship between the inputs and outputs. This equation may come from process knowledge (the length of a project is just the sum of each phase), or from a statistical analysis where you fit a predictive model such as a designed experiment (DOE) or regression analysis on the historical data in Minitab.

A Monte Carlo simulation also depends on a reasonable specification of the distribution for each input, which defines the variation. If you do not know which distribution to use, Companion can examine historical data in a CSV file and recommend a possible distribution.

Each iteration of the simulation draws a random sample of possible values from the distributions for each input, enters these values into the equation, and then calculates the outputs. The number of iterations should adequately cover the range of possible input and output values, and provide accurate results. The maximum number is 1,000,000. However, the default of 50,000 is sufficient for most models.

There is no limit to the number of inputs and outputs in the model. Your simulation can be as large as necessary.

## How-to

1. Identify the equations.
2. Define the distribution of each input variable.
3. Add a Monte Carlo simulation.
4. Perform a parameter optimization.
5. Perform a sensitivity analysis.