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# Design of Experiments (DOE) Tutorial

Design of experiments (DOE) is a powerful tool that can be used in a variety of experimental situations. DOE allows for multiple input factors to be manipulated determining their effect on a desired output (response). By manipulating multiple inputs at the same time, DOE can identify important interactions that may be missed when experimenting with one factor at a time. All possible combinations can be investigated (full factorial) or only a portion of the possible combinations (fractional factorial). Fractional factorials will not be discussed here.

When to Use DOE

Use DOE when more than one input factor is suspected of influencing an output. For example, it may be desirable to understand the effect of temperature and pressure on the strength of a glue bond.

DOE can also be used to confirm suspected input/output relationships and to develop a predictive equation suitable for performing what-if analysis.

DOE Procedure

1. Acquire a full understanding of the inputs and outputs being investigated. A process flow diagram or process map can be helpful. Utilize subject matter experts as necessary.
2. Determine the appropriate measure for the output. A variable measure is preferable. Attribute measures (pass/fail) should be avoided. Ensure the measurement system is stable and repeatable.
3. Create a design matrix for the factors being investigated. The design matrix will show all possible combinations of high and low levels for each input factor. These high and low levels can be generically coded as +1 and -1. For example, a 2 factor experiment will require 4 experimental runs:
 Input A Level Input B Level Experiment #1 -1 -1 Experiment #2 -1 +1 Experiment #3 +1 -1 Experiment #4 +1 +1

Note: The required number of experimental runs can be calculated using the formula 2n where n is the number of factors.

4. For each input, determine the extreme but realistic high and low levels you wish to investigate. In some cases the extreme levels may be beyond what is currently in use. The extreme levels selected should be realistic, not absurd. For example:

 -1 Level +1 Level Temperature 100 degrees 200 degrees Pressure 50 psi 100 psi

5. Enter the factors and levels for the experiment into the design matrix. Perform each experiment and record the results. For example:

 Temperature Pressure Strength Experiment #1 100 degrees 50 psi 21 lbs Experiment #2 100 degrees 100 psi 42 lbs Experiment #3 200 degrees 50 psi 51 lbs Experiment #4 200 degrees 100 psi 57 lbs

6. Calculate the effect of a factor by averaging the data collected at the low level and subtracting it from the average of the data collected at the high level. For example:

Effect of Temperature on strength:
(51 + 57)/2 - (21 + 42)/2 = 22.5 lbs

Effect of Pressure on strength:
(42 + 57)/2 - (21 + 51)/2 = 13.5 lbs

7. The interaction between two factors can be calculated in the same fashion. First, the design matrix must be amended to show the high and low levels of the interaction. The levels are calculated by multiplying the coded levels for the input factors acting in the interaction. For example:

 Input A Level Input B Level Interaction Experiment #1 -1 -1 +1 Experiment #2 -1 +1 -1 Experiment #3 +1 -1 -1 Experiment #4 +1 +1 +1
8. Calculate the effect of the interaction as before. Effect of the interaction on strength: (21 + 57)/2 - (42 + 51)/2 = -7.5 lbs
9. The experimental data can be plotted in a 3D Bar Chart.

The effect of each factor can be plotted in a Pareto Chart.

The negative effect of the interaction is most easily seen when the pressure is set to 50 psi and Temperature is set to 100 degrees. Keeping the temperature at 200 degrees will avoid the negative effect of the interaction and help ensure a strong glue bond.

### Conduct and Analyze Your Own DOE

Conduct and analyze up to three factors and their interactions by downloading the 3-factor DOE template (Excel, 104 KB).

### Summary

More complex studies can be performed with DOE. The above 2-factor example is used for illustrative purposes. A thorough discussion of DOE can be found in Juran’s Quality Handbook.

Contributed by Dean Christolear.