Response Surface DOE

Summary

Provides a cost-effective methodology for conducting controlled experiments (DOEs) in cases where there is believed to be curvature and all the factors are continuous and can be tested at (usually) three to five levels. The goals of this type of experiment are usually focused on developing a full predictive model (Y = f(X)) describing how the process inputs jointly affect the process output and determining the optimal settings of the inputs.

Answers the questions:
  • Which process inputs (factors) have the largest effects on the process output (which inputs are the key inputs)?
  • Do important interactions exist between factors?
  • Do important quadratic effects exist?
  • How much of the variation in the process output can be explained by varying the process inputs?
  • Is the current testing space near an optimal condition for the process output? If yes, what settings of the key inputs result in the optimal process output?
  • What is the equation (Y = f(X)) relating the process output to the levels of the factors?
When to Use Purpose
Mid-project If the number of process inputs to be investigated is small (typically less than seven), you can run these designs by adding new test runs to an existing 2-level full or fractional factorial design when the 2-level factorial design shows evidence of curvature. All factors must be continuous.
Mid-project When all the factors are continuous and show significant curvature, these designs are used because they allow the fitting of quadratic terms to model the curvature, resulting in better interpolation between design points and an improved search for the optimal settings.

Data

Continuous Y, continuous X's tested at three to five discrete levels.

How-To

  1. Generally, you have already performed a series of DOEs and simple point evaluations to arrive at the general area of optimization. This is sometimes called response surface methodology or path of steepest ascent.
  2. Once here, you add star points (CCD method) to a 2k full or 2k fractional (Res V+) design.
  3. Run the additional points and reduce to a final model by eliminating terms with high p-values (typically greater than 0.05).
  4. Use the response optimizer or the surface and contour plots to determine settings of final factors.
  5. Generate the prediction equation.

Guidelines

  • First, you should develop a sound data collection strategy so that your conclusions are based on truly representative data.
  • Whenever possible, you should do the runs in the experiment in random order to prevent confusing a factor effect with the effect of an untested factor (sometimes called a lurking variable).
  • The residuals of the final model must be independent, be reasonably normal, and have reasonably equal variance. The residuals are usually analyzed by a histogram, normal probability plot, residuals versus fits, and residuals versus order, which you can run at one time using the Four in one option. Note: Due to the small size of many DOEs, these assumptions may not be easily checked.
  • Minitab supports both central composite designs and Box-Behnken designs. Central composite designs are often preferred over the Box-Behnken designs because they can usually be built on prior 2k full or fractional (resolution V or higher) DOEs.
  • Check for possible outliers in the unusual observations table (Session window output).
  • Do not extrapolate beyond your inference space.
  • While the discussion here focuses on designed experiments created by Minitab, note that you can use the Analyze portion of the response surface DOE to analyze any numeric experimental data (for example, two factors each at 10 levels). To do this, enter the Y and X data into Minitab and then use Stat > DOE > Response Surface > Define Custom Response Surface Design to define the factors. You can analyze this newly defined custom design in the usual manner.
  • If you have discrete numeric data from which you can obtain every equally spaced value and you have measured at least 10 possible values, you can evaluate these data as if they are continuous.
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