Response Surface Methodology

Response surface methodology

• RSM can be defines as a statistical method that uses quantitative data from appropriate experiments to determine & simultaneously solve multivarient equations

Factors to consider

• Critical factors are known - system well understood
• Region of interest , where factor levels influencing product is known
• Factors vary continuously through- out the experimental range tested
• A mathematical function relates the factors to the measured response
• The response defined by the function is a smooth curve

Limitations to RSM

• Large variations in the factors can be misleading (error, bias, no replication)
• Critical factors may not be correctly defined or specified
• Range of levels of factors to narrow or to wide --optimum can not be defined
• Response not linear
• Lack of use of good statistical principles
• Over-reliance on computer -- make sure the results make good sense

Uses of RSM

• To determine the factor levels that will simultaneously satisfy a set of desired specifications
• To determine the optimum combination of factors that yield a desired response and describes the response near the optimum
• To determine how a specific response is affected by changes in the level of the factors over the specified levels of interest
• To achieve a quantitative understanding of the system behavior over the region tested
• To product product properties throughout the region - even at factor combinations not actually run
• To find conditions for process stability = insensitive spot

Process models

• Ym = fm(x1, x2, ….,xp)
• Polynomials with a small number of terms are most desirable
• Most process outputs are some sort of smooth function of the inputs
• Second-degree polynomials are generally adequate

Polynomial models

• Second degree - one independent variable

• For p factors, there will be one constant term, p linear terms

• Polynomial model does a poor job of predicting response outside the region of experimentation

Designs

• Predictions always have some degree of uncertainty
• Should have reasonable prediction throughout the experimental range
• Uniform predictions error is obtained by using a design the fills out the region of interest
• The choice of experimental design is affected by the shape of the experimental region
• In most cases, the region is determined by the ranges of the independent variable. In this case the region is cubical (in coded values of x) and the best design in face centered
• If "standing the the center" and one it is desired that the precision of predications be independent of direction from center -then the region is spherical and design of choice is Box-Behnken
• Box-Behnken designs exclude the corners, where all variable are simultaneously at the maximum levels - therefore Box-Behnken design permits a wider range of individual ranges.
• If the shape of the experiment is neither spherical or cubical and has strong constraints - then the region may be an irregular tetrahedron and will require a special design

Face centered cube

• Two-level factorial
• Two face centered points for each factor
• Three or more center points
• When run in blocks, center points are run with each block
• Face points are runs for which all factors except one are at the middle setting - and provide the information needed to determine curvature

Blocking

• In large sizes, both face-centered cube and box-behnken permit blocking.
• Difference (or biases) in the level of the responses between blocks with not affect estimates of coefficients nor estimates of the factor and interaction effects

Face centered cube

• The main part of the face-centered cube design is a two-level factorial, which fills out a cubic region
• The face points constitute a separate block - so that the first two blocks, which comprise a two level factorial, can be run first.
• The face points are added if serious curvature is found
• "piggy back" approach gives flexibility
• Center points are need to provide good predictors of center of region
• For 3 or more factors, it is best to use blocks

Box-behnken design

• The box-behnken design fills out a polyhedron, approximating a sphere
• For 3 factors (15 runs) the design consist of three four-run, two-level factorials in two factors, with the third factor at its mid-level and three center point - run in three blocks of 10 runs
• For a 3 factor experiment, the 15 runs consist of three four-run, two-level factorials in two factors - with the third factor at its mid-level, and three center points.
• Box-behnken and face-centered cubic designs are subsets of the full three level factorial designs. Except for center points, they are complementary fractions in that no point in one design is in the other design

Design choice

• Face centered cube and Box-Behnken take about the same number of experiments
• If time or money dictates fewer that the required number of independent variables, then consider -

Operability review

• Runs should be reviewed for operability.
• Runs that set all the "driving force" variables at minimum or maximum values may not work
• Randomization can be altered to schedule these runs early to allow for latter adjustments
• Exploratory testing of potential troublesome runs before experimentation should be considered
• You may find, part-way through the experiment that some design points will not run. This is true is a boundary curve passes through the experimental region.
• If only one or a few points are involved, they may be moved towards the center, just enough to become operable
• All standard response surface designs are robust against modest displacement or a few data points

Avoiding blunders

• Execute experiment with care. Small statistical designs are susceptible to errors because every run estimates more than one effect
• Record results for all runs
• Plan for analysis from the beginning
• A computer is generally required for analysis - and regression analysis is the basis for most analytical procedures
• Make sure the results "make sense"

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