## Key Points

- Full factorial DOE is one of several methods of approaching experimental design.
- Full factorial DOE allows you to control the main effects of the experiment.
- Utilizing this method allows you to filter through the noise to focus on what matters.
- You might have to utilize screen experiments to make the most of full factorial DOE.

Full factorial DOE, or Design of Experiments, is a method of designed experimentation. Using this method you manipulate the controllable factors (independent variables or inputs) in your process at different levels to see their effect on some response variable (dependent variable or output).

This article will explore the different approaches to DOE with a specific focus on the full factorial design. We will discuss the benefits of using the full factorial design and offer some best practices for a successful experiment.

## Overview: What is a Full Factorial DOE?

As stated above, a full factorial DOE design is one of several approaches to designing and carrying out an experiment to determine the effect that various levels of your inputs will have on your outputs. The purpose of the DOE is to determine at what levels of the inputs will you optimize your outputs.

For example, if your output is the thickness of the coating to be applied to a metal sheet. Your primary process variables are speed, temperature, and viscosity of the coating. What combination of speed, temperature, and viscosity should you use to get an optimal and consistent thickness on the metal sheet?

With three variables, speed, temperature, and viscosity, how many different unique combinations would we have to try to fully examine all the possibilities? Which combination of speed, temperature, and viscosity will give us the best coating thickness? The experimentation using all possible factor combinations is called a full factorial design. The minimum number of experiments you would have to do is called *Runs*.

We can calculate the total number of unique factor combinations with the simple formula of *# Runs=2^k*.

*k* is the number of variables and *2 *is the number of levels, such as (high/low) or (400 rpm/800 rpm). In our coating example, we would call this design a *2-level*, *3-factor** full factorial DOE*. The number of runs would then be calculated as 2^3, or 2x2x2, which equals 8 total runs.

There are other designs that you can use such as a fractional factorial. This uses only a fraction of the total runs. That fraction can be one-half, one-quarter, one-eighth, and so forth depending on the number of factors or variables.

## Benefits of Doing a Full Factorial DOE

Doing a full factorial as opposed to a fractional factorial or other screening design has several benefits.

### You Can Determine the Main Effects

Main effects describe the impact of each factor on the output or response variable. In our example, one of the main effects would be the impact or change in the coating thickness. This would be attributable to speed alone if we changed from a run speed of 400 rpm to a speed of 800 rpm.

### You Can Determine the Effects of Interactions on the Response Variable

An interaction occurs when the effect of one factor on the response depends on the setting of another factor. For our example, if we ran at a speed of 800 rpm, what temperature should we run at to optimize our coating thickness? On the other hand, what temperature should we run at if the speed is 400 rpm?

### The Optimal Settings for the Independent Variables Can Be Estimated

The final full factorial analysis will tell us what setting or levels of our speed, temperature, and viscosity we should use to optimize our coating thickness.

Our example answer might look like this: Run the machine at 400 rpm, a temperature of 350 degrees using a coating viscosity of 6,000 cps.

## Why Is a Full Factorial DOE Important to Understand?

Using intuition to set the optimal settings of your process variables or factors is insufficient if the goal is to understand the impact of the factors on your output. Using trial and error may miss the important combinations or optimal combinations, and you might end up with a less-than-optimized process or product.

### You Need to Know the Full Effect of Your Variables on the Process

Your process variables have different impacts on your output. You need the whole picture.

### The World Is Not Just Made Up of Main Effects

As explained earlier, the main effects are the individual impacts of each factor on the output. But the world is more complex than that, and most outcomes are a function of interactions, not just main effects.

### State Your Conclusions with Statistical Certainty

You can determine the main effects, interactions, and other outcomes of a full factorial DOE using statistics. As such, decisions are based on statistical significance rather than hunches or “seat of the pants” conclusions.

## An Industry Example of a Full Factorial DOE

A beverage manufacturer wanted to reduce the amount of overfilled bottles on its manufacturing line. Company leadership felt that the major factors in the process were the run speed of the machine, size of the bottle, type of product, and degree of carbonation. Therefore, a 2-level, 4-factor full factorial experiment was selected. This would require 16 runs.

The company’s Master Black Belt designed the DOE and ran it on a preselected machine. Accordingly, the experiment was restricted to a single machine to block out any impact that might be attributable to machine differences.

Each run consisted of 100 bottles. Further, they took the average of the fill level of those 100 to use in the calculations. Accordingly, running a single bottle would have been impractical. They determined that, after doing the calculations and analysis all four factors were statistically significant — and that there were interactions between speed and carbonation. As such, the optimal settings suggested by the experiment were used in a confirmatory run to see if the changes improved the fill level.

It did, and they replicated the new settings on the rest of the machines.

## Why Does It Matter?

We’ve discussed a lot of the how behind full factorial DOE, but not a lot of the why. So, why should you use full factorial DOE when conducting experimentation? The pure and simple reason is that it gives you total control over the constraints and criteria specified for an experiment. When testing anything, you want a controlled environment, and full factorial DOE allows for that.

## Best Practices When Thinking About a Full Factorial DOE

Running a DOE needs planning and discipline. The results of your experiment can become contaminated and untrustworthy if you don’t take care.

### Clearly Define Your Factors and Desired Outcomes

Know what factors are likely to be the most important and how you are going to measure them.

### Minimize the “Noise”

Since you’re only interested in the impact of your chosen factors on the response, remove any other factors that might contaminate your experiments. Control and minimize any other factors around you so that they don’t inadvertently affect your experiment.

### Use Screening Experiments if Appropriate

If you have a large number of possible factors, you will be doing a large number of runs that can get costly and time-consuming. To help screen out the factors that are not important, use appropriate screening experiments such as a fractional factorial.

## Other Useful Tools

Full factorial DOE isn’t the only tool at your disposal. When undertaking any sort of analysis, it helps to be fully prepared. As such, you might want to consider learning the ins and outs of upper control limits. This handy metric allows you to see when things are going too far out of spec.

Additionally, if you’re stuck on a decision, there are a few different strategies to consider. One of the most powerful tools at your disposal is the Pugh Matrix. This concept allows for nimble decision-making, allowing it to become an indispensable part of your toolbox.

## In Summary: Full Factorial Doe

Designed experiments in general and a full factorial DOE design in particular, are powerful statistical tools to understand your process and optimize your output. However, you must take care to do the DOE with planning and discipline so the results are meaningful.

In a full factorial DOE, you will identify the appropriate output that you want to improve and the factors or variables that you believe impact that output. Once you’ve identified the factors, determine the levels or settings you’d like to explore and the number of unique combinations of the factors and levels.

After running your experiment, you’ll usually use a statistical software package to analyze your results. From there, you will be able to statistically determine the main effects of your factors, the interactions between the factors, and the optimal levels or settings.