Engineering Statistics ENGR 592 Prepared by: Mariam El-Maghraby Date: 26/05/04 Design of Experiments Plackett-Burman Box-Behnken

Presentation Outline I. Introduction to ‘Fractional factorial designs’ II. Plackett-Burman A. Assumptions and main properties B.When to use PB designs C. Example III. Box-Behnken A. Assumptions and main properties B. When to use Box-Behnken

I. Fractional Factorial Designs As k increases, the runs specified for a 2 k or 3 k full factorial quickly become very large and outgrow the resources of most experimenters Solution: to assume that certain high-order interactions are negligible by using only a fraction of the runs specified by the full factorial design Properly chosen fractional factorial designs for two-level experiments have the desirable properties: – Balanced: All runs have the same number of observations) – Orthogonal: Effects of any factor sum to zero across the effects of the other factors).

II. Plackett-Burman A. Properties & Assumptions : Fractional factorial designs for studying k = N – 1 variables in N runs, where N is a multiple of 4. Only main effects are of interest. Standard orthogonal arrays. No defining relation since interactions are not identically equal to main effects. All information is used to estimate the parameters leaving no degrees of freedom to estimate the error term for the ANOVA. Plackett and Burman (1946) showed how full factorial designs can be fractionalized in a different manner than traditional 2 k-p fractional factorial designs, in order to screen the max number of (main) effects in the least number of experimental runs

II. Plackett-Burman B. When to use PB designs: Screening Possible to neglect higher order interactions 2-level multi-factor experiments. More than 4 factors, since for 2 to 4 variables a full factorial can be performed. To economically detect large main effects. Particularly useful for N = 12, 20, 24, 28 and 36.

II. Plackett-Burman C. Example: Experiment to study the eye focus time Response: Eye Focus time 1.Factors: 1.Sharpness of vision 2.Distance from target to eye 3.Target shape 4.Illumination level 5.Target size 6.Target density 7.Subject. Screening experiment Two levels of each factors are considered while higher order interactions are negligible Required to economically identify the most important factors

III. Box-Behnken Box-Behnken designs are the equivalent of Plackett-Burman designs for the case of 3**(k-p). A.Properties & Assumptions: Three-level multi-factor experiments. Combine two-level factorial designs with incomplete block designs. Complex confounding of interaction. Economical Fill out a polyhedron approximating a sphere. Can test the linear and quadratic (non-linear) effect for each factor. Standard orthogonal arrays. The treatment combinations are at the midpoints of edges of the process space and at the center. Rotatable

Number of runs required by Central Composite and Box-Behnken designs Number of factors Box BehnkenCentral Composite 2-13 (5 center points) (6 center point runs) (6 center point runs) (fractional factorial) or 52 (full factorial) (fractional factorial) or 91 (full factorial) [source: Engineering Statistics Handbook]

III. Box-Behnken B. When to use Box-Behnken designs: With three-level multi-factor experiments. When it is required to economically detect large main effects, especially when it is expensive to perform all the necessary runs. When it is required to determine the linear and quadratic effects of each variable. When the experimenter should avoid combined factor extremes. This property prevents a potential loss of data in those cases.

III. Box-Behnken C. Example: Experiment to study the yield of a chemical process Response: Chemical process yield Factors: 1.Temperature 2.pH 3.Pressure 4.Viscosity 5.Mixer speed Screening experiment Three levels of each factors are considered while higher order interactions are negligible Required to economically identify the most important factors

Summary of desirable features of Box-Behnken designs Satisfactory distribution of information across the experimental region (rotatability) Fitted values are as close as possible to observed values (minimize residuals or error of prediction) Good lack of fit detection. Internal estimate of error. Constant variance check. Transformations can be estimated. Suitability for blocking. Sequential construction of higher order designs from simpler designs Minimum number of treatment combinations. Good graphical analysis through simple data patterns. Good behavior when errors in settings of input variables occur.

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