Transcript Document

Engineering Statistics
ENGR 592
Design of Experiments
Plackett-Burman
Box-Behnken
Prepared by: Mariam El-Maghraby
Date: 26/05/04
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 2k or 3k 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
Plackett and Burman (1946) showed how full
factorial designs can be fractionalized in a different
manner than traditional 2k-p fractional factorial
designs, in order to screen the max number of (main)
effects in the least number of experimental runs
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.
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.
2.
3.
4.
5.
6.
7.
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Sharpness of vision
Distance from target to eye
Target shape
Illumination level
Target size
Target density
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:
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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 Behnken
Central Composite
2
-
13 (5 center points)
3
15
20 (6 center point runs)
4
27
30 (6 center point runs)
5
46
33 (fractional factorial) or 52
(full factorial)
6
54
54 (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
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Response: Chemical process yield
Factors:
1.
2.
3.
4.
5.
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Temperature
pH
Pressure
Viscosity
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
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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.
Thank you
Questions