- Design of experiments (DOE) is a systematic, rigorous problem-solving approach to provide valid conclusions.
- minimal expenditure of engineering runs, time, and money.
The 5 problem areas in which DOE may be applied:
- whether or not a particular factor is “significant”(effect of change in a single factor on overall process)
- Screening/Characterizing (main effects designs)-
- screen out the few important main effects from the many less important ones.
- understand the process (most important to least important factors)
- Response Surface (method) RSM objective–
- allows to estimate interaction and even quadratic effects,
- and therefore give an idea of the (local) shape of response surface we are investigating
- Make a product or process more robust against external and non-controllable influences.
- Optimizing responses when factors are proportions of a mixture (mixture design):-“best” proportions/levels of the factors to maximize (or minimize) a response.
- Optimal fitting of a regression model (regression design):-modeling a response/output as a mathematical function (either known or empirical) of a few continuous(numerical) factors and we desire “good” model parameter estimates (i.e., unbiased and minimum variance), (high predictive power).
Centerpoints and Axial points(Star points)
- We add centerpoint runs interspersed among the experimental setting runs for two purposes: 1)To provide a measure of process stability and inherent variability. 2) To check for curvature.
- Center points are simply experimental runs where predictor variables X’s are set halfway between (i.e., in the center of) the low and high settings.
- Centerpoint runs should begin and end the experiment, and should be dispersed as evenly as possible throughout the design matrix.
- Star points (Axial points) establish new extremes for high and lowlevels for all factors.
- Best overall design performance occurs with α ≈ √k and 2 ≤ nc ≤5 (number of center points (nc) for given k factors).
- Alpha (α) is the distance of each axial point (also called star point) from the center in a central composite design.
- α value less than one puts the axial points in the cube;
- α value equal to one puts them on the faces of the cube;
- α value greater than one puts them outside the cube.
- Alpha, along with the number of center points, determines whether a design can be orthogonally blocked and is rotatable.
- An experimental design is orthogonal if each factor can be evaluated independently of all the other factors.
- Uniformity of prediction error.
- A design is rotatable if the variance of the predicted response at any point x depends only on the distance of x from the design center point.
- Rotatability is a desirable property for response surface designs (i.e. quadratic model designs).
- Resolution describes the degree to which estimated main effects re confounded (or aliased) with estimated 2-level interactions, 3-level interactions, etc.
- In general, the resolution of a design is one more than the smallest order interaction that some main effect is confounded (aliased) with. If some main effects are confounded with some 2-level interactions, the resolution is 3.
- Full factorial designs have no confounding and are said to have resolution “infinity”.
- For most practical purposes, a resolution 5 design is excellent and a resolution 4 design may be adequate. Resolution 3 designs are useful as economical screening designs.
- Factorial designs assume there’s a linear relationship between each X and Y. Therefore, if the relationship between any X and Y exhibits curvature, we shouldn’t use a factorial design because the results may be misleading.
- Center points can help statistically determine if the relationship is linear or not? If the center point p-value is significant (i.e., less than alpha), then we can conclude that curvature exists and should use response surface DOE—such as a central composite design or Box Behnken design to analyze data.
- Crossed Factors: Two factors are crossed if every level of one occurs with every level of the other in the experiment.
- Nested Factors: A factor “A” is nested within another factor “B” if the levels of “A” are different for every level of “B”.
- Lack of Fit Error: occurs when the analysis omits one or more important terms or factors from the process model.
- Random error (experimental error): occurs due to natural variation in the process.
- Contour plot: explore the relationship between three variables-two factors (predictor variables) plotted on X and Y axes and one response variable plotted as contours (3-D relation is shown in 2D). These contours are sometimes called z-slices or iso-response values. Different colors show different countours corresponding to different X & Y factor values. We can select best output matching our ‘set objective’.
- Surface plot: is a companion plot to the contour plot. Rather than showing the individual data points, surface plot shows a functional relationship between two independent variables (X and Z) plotted on horizontal axes and a dependent variable (Y) plotted on vertical axis. We can rotate the plot to different angles to verify the conclusion (Highest or lowest vertical points). https://www.youtube.com/watch?v=9YNcvC834rc
- Designs are a collection of rows and columns that define the framework of the experiment.
- The rows are the runs. The columns contain design information, components, factors, and responses.
- There are four main study types:
- Factorial design
- primarily used for understanding if factors are important to the process; either for screening of few important factors out of many possibilities, or characterizing how known factors interact and individually effect the process.
- often used as a starting point for more complex response surface modeling.
- Response surface Methodology (RSM)
- Mixture design
- Mixture designs are used when the response changes as a function of the relative proportions of the components.
- All components must be entered in the same units of measure and each run must sum to the same total.
- Custom design
- when the process requires adjustments to the experiment that cannot be accommodated by a standard design.
Response surface Methodology (RSM)
- Robust mapping of response.
- Difference between a RSM equation and the equation for a factorial design is the addition of squared (or quadratic) terms that lets us model curvature in the response.
- Most efficient approach would be to first construct a Factorial deisign to test if a first order model is sufficient? If curvature is present or optimization is required then a CCD could be run to obtain a second order model.
- Higher order of polyomials provide higher precision.
- There are two main types of response surface designs:Box-Behnken designs and Central composite designs.
- CCD are based on 2-level factorial designs, augmented with center and axial points to fit quadratic models. Regular CCD’s have 5 levels for each factor. This can be modified by choosing an axial distance of 1.0 creating a Face-Centered, Central Composite design which has only 3 levels per factor.
- Box-Behnken designs always have three levels for each factor and are purpose built to fit a quadratic model. The Box-Behnken design does not have runs at the extreme combinations of all the factors, but compensates by having better prediction precision in the center of the factor space.Box-Behnken designs usually have fewer design points than central composite designs, thus, they are less expensive to run with the same number of factors.They can efficiently estimate the first- and second-order coefficients; however, they can’t include runs from a factorial experiment.A Box-Behnken design also requires only three-levels, and is a more efficient alternative to the full three-level factorial. BBD avoids all the corner points and star points.
Response Surface Design
- A response surface is a geometrical representation of a response variable plotted as a function of the independent variables.
- The Central-Composite designs build upon the two-level factorial designs by adding a few center points and axial (star) points. The CC designs (Box and Wilson designs) are constituted of a full factorial or fractional design. The points at the center of the experimental domain and the “star” points outside this domain make it possible to estimate the curvature of the response surface.
- The number of experiments to perform in a centered composite design is determined by the following formula when the factorial design is full: N = 2k+2k+N0. (k=number of factors, N0=number of points at center)
- A factor’s five values are: -a, -1, 0, 1, and a. The value of a is determined by the number of factors in such a way that the resulting design is orthogonal. For example, if you are going to use either four or five factors, the value of a is 2.00.
- The actual values of the levels are determined from these five values as follows:
- The low-level value is assigned to -1.
- The high-level value is assigned to 1.
- The average of these two values is assigned to 0.
- The values of -a and a are used to find the minimum and the maximum values.
- The diagrams below illustrate the three types of central composite designs for two factors.
- Note that the CCC explores the largest process space and the CCI explores the smallest process space.
- Both the CCC and CCI are rotable designs, but the CCF is not.
- In the CCC design, the design points describe a circle circumscribed about the factorial square. For three factors, the CCC design points describe a sphere around the factorial cube.
- The Central Composite design is based on a two-level factorial design with the addition of 2k (k is the number of independent variables) points (star points) between the axes plus repeat points at the centroid.