## Simulation Models

A **simulation model** is a mathematical model that calculates the impact of uncertain inputs and decisions we make on outcomes that we care about, such as profit and loss, investment returns, environmental consequences, and the like. Such a model can be created by writing code in a programming language, statements in a simulation modeling language, or formulas in a Microsoft Excel spreadsheet. Regardless of how it is expressed, a simulation model will include:

**Model**that are uncertain numbers -- we'll call these uncertain variables*inputs*- Intermediate calculations as required
**Model**that depend on the inputs -- we'll call these uncertain functions*outputs*

It's essential to realize that model *outputs* that depend on uncertain *inputs* are uncertain themselves -- hence we talk about **uncertain variables** and **uncertain functions**. When we perform a simulation with this model, we will test many different numeric values for the uncertain variables, and we'll obtain many different numeric values for the uncertain functions. We'll use **statistics** to analyze and summarize all the values for the uncertain functions (and, if we wish, the uncertain variables).

#### Creating Models in Excel or Custom Programs

An **Excel spreadsheet** can be a simple, yet powerful tool for creating your model -- especially when paired with Monte Carlo simulation software such as **Risk Solver**. If your model is written in a **programming language**, Monte Carlo simulation toolkits like the one in Frontline's **Solver Platform SDK** provide powerful aids.

An example model in Excel might look like this, where cell B6 contains a formula =PsiTriangular(E9,G9,F9) to sample values for the uncertain variable Unit Cost, and cell B10 contains a formula =PsiMean(B9) to obtain the mean value of Net Profit across all trials of the simulation.

A portion of an example model in the C# programming language might look like this, where the array Var[] receives sample values for the two uncertain variables X and Y, and the uncertain function values are computed and assigned to the Problem's FcnUncertain object Value property:

#### Choosing Samples for Uncertain Variables

We must also choose what **random sample values** to use for the uncertain variables. During a simulation, a new sample value will be drawn for every uncertain variable on each trial. **Risk Solver** provides state-of-the-art random number generators and sampling methods for your simulation needs.

In the simplest case, we might generate **random numbers** between 0 and 1, and use these as sample values. But in most cases, the **range** of values, and chance that different values in the range will be drawn on each trial, must be tailored to the uncertain variable. To do this, we normally choose a **probability distribution** and appropriate **parameters** for the uncertain variable. As discussed next, selecting appropriate probability distributions is a **key step** in building a simulation model.

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