Design of experiments-Box behnken design

Design of
Experiments
Box Behnken Design
Dr. KOMAL G DAVE
Associate professor
L.D. College of Engineering, Ahmedabad, Gujarat.
GUIDED BY:-
GULAMHUSHEN SIPAI
160280708016
PREPARED BY:-

WHAT IS EXPERIMENT?
In statistics, an experiment refers to any process that
generates a set of data.
An experiment involves a test or series of test in which purposeful
changes are made to the input variables of a process or system so that
changes in the output responses can be observed and identified.
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What is DOE
Design of Experiment (DOE) is a powerful statistical technique for
improving product/process designs and solving process /
production Problems
When analyzing a process, experiments are often used to evaluate
which process inputs have a significant impact on the process
output and what the target level the inputs should be to achieve a
desired result (output).
Design of Experiments (DOE) is also referred to as Designed
Experiments or Experimental Design
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Why DOE ?
Reduce time to design/develop new products & processes
Improve performance of existing processes
Improve reliability and performance of products
Achieve product & process robustness

DOE EXAMPLE

TERMINOLOGIES
RESPONSE
FACTORS
LEVELS
NOISE
Factors are those quantities that affect the
outcome of an experiment.
A measurable outcome of interest
Values that factor will take in an experiment
Variables that affect product / process performance,
whose values can not be controlled or are not
controlled for economic reasons.

TERMINOLOGIES
INTERACTION • Sometimes factors do not behave the same when they are looked at together as when they
are alone; this is called an interaction
• Interaction plot can be used to visualize possible interactions between two or more factors
• Parallel lines in an interaction plot indicate no interaction
• The greater the difference in slope between the lines, the higher the degree of interaction
Randomization
• Randomization is a statistical tool used to minimize impact of noise in the experiment by
randomly assigning material, people, order that experimental trials are conducted, or any
other factor which are not under the control of the experimenter
BLOCKING
• Blocking is a technique used to increase the precision of an experiment by breaking the
experiment into homogeneous segments in order to control any potential block to block
variability

TERMINOLOGIES
RAPLICATION
• Replication is making multiple experimental runs for each experiment combination.
• Repetition of a basic experiment without changing any factor settings, allows the
experimenter to estimate the experimental error (noise) in the system

EXPERIMENT DESIGN PROCESS

02
03
01
TYPES OF DOE
One Factorial
Full Factorials
Fractional Factorials
Screening Experiments
Response Surface Analysis
05
04

ONE FACTORIAL METHOD
One factorial experiments look at only one
factor having an impact on output at
different factor levels.
In single factor experiments, ANOVA models
are used to compare the mean response
values at different levels of the factor.
Each level of the factor is investigated to see
if the response is significantly different from
the response at other levels of the factor.
The analysis of single factor experiments is
often referred to as one-way ANOVA

FULL FACTORIAL METHOD
Full factorial experiments look completely at all
factors included in the experimentation.
The effects that the main factors and all the
interactions between factors are measured.
If we use more than two levels for each factor, we
can also study whether the effect on the
response is linear or if there is curvature in the
experimental region for each factor and for the
interactions.

FRACTIONAL FACTORIAL METHOD
Fractional factorials look at more factors with
fewer runs.
Using a fractional factorial involves making a
major assumption – that higher order
interactions (those between three or more
factors) are not significant.
To increase the efficiency of experimentation,
fractional factorials give up some power in
analyzing the effects on the response. Fractional
factorials will still look at the main factor effects,
but they lead to compromises when looking into
interaction effects. This compromise is called
confounding.

SCREENING EXPERIMENTS
Screening experiments are the ultimate fractional factorial experiments. These experiments assume that
all interactions, even two-way interactions, are not significant.
They literally screen the factors, or variables, in the process and determine which are the critical variables
that affect the process output.
There are two major families of screening experiments:
1. Drs. Plackett and Burman developed the original family of screening experiments matrices in the 1940s.
2. Dr. Taguchi adapted the Plackett–Burman screening designs. He modified the Plackett–Burman design
approach so that the experimenter could assume that interactions are not significant, yet could test for
some two-way interactions at the same time.

Response surface methodology
Response surface methodology (RSM) is a collection of mathematical and statistical techniques for model
building.
Originally, RSM was developed to model experimental responses (Box and Draper, 1987), and then
migrated into the modelling of numerical experiments.
y = f (x1, x2) + e
In RSM the objective of DoE is the selection of the points where the response should be evaluated.
𝟑 𝟑 full factorial design (27 points)
Three one-third fractions of the 𝟑 𝟑 design

Central composite design for
3 design variables at 2 levels
CCD D-optimal designs
Y = X * B + e
B = (𝑋 𝑇
∗ 𝑋 )−1
𝑋 𝑇
Y
Latin hypercube design
This method ensures that every variable is
represented, no matter if the response is dominated by
only a few ones. Another advantage is that the number
of points to be analyzed can be directly defined.
--Schoofs et al. (1997)-Giunta et al. (1996)
-Haftka and Scott (1996)
Audze-Eglais' approach
Audze and Eglais (1977) suggested a non-traditional criterion for elaboration of plans of experiments which, similar
to the Latin hypercube design, is not dependent on the mathematical model of the problem under consideration. The
input data for the elaboration of the plan only include the number of factors N (number of design variables) and the
number of experiments K.
Eschenauer and
Mistree (1997)

Comparison between CCD (a), Latin hypercube design (b) and Audze-Eglais design (c)

The Box–Behnken Design.
A Box–Behnken design for three factors
A face-centered central composite design for k 3

Annadurai, G., and R. Y. Sheeja. "Use of Box-Behnken design of experiments for the adsorption of verofix red
using biopolymer." Bioprocess engineering 18.6 (1998): 463-466.

Ferreira, SL Costa, et al. "Box-Behnken design: an alternative for the optimization of analytical methods.
" Analytica chimica acta 597.2 (2007): 179-186.
Applications:
1. optimization of the spectro analytical method
2. optimization of chromatographic methods
3. optimization of capillary electrophoresis
4. optimization of electro analytical methods
5. optimization of sorption process
BBD permits:
(i) estimation of the parameters of the quadratic
model;
(ii) building of sequential designs;
(iii) detection of lack of fit of the model; and
(iv) use of blocks.

Aslan, N., and Y. Cebeci. "Application of Box–Behnken design and response surface methodology for modeling of
some Turkish coals." Fuel 86.1-2 (2007): 90-97.

References
1. Montgomery, Douglas C. Design and analysis of experiments. John wiley & sons, 2017
2. Box, George EP, and Norman R. Draper. Empirical model-building and response surfaces. John Wiley & Sons, 1987.
3. Lautenschlager, Uwe, Hans A. Eschenauer, and Farrokh Mistree. "Multiobjective flywheel design: a doe-based concept
exploration task." Advances in Design Automation (Dutta, D., ed.) (1997): 14-17.
4. Giunta, Anthony A., et al. "Wing design for a high-speed civil transport using a design of experiments methodology."
(1996).
5. Haftka, Raphael T., Elaine P. Scott, and Juan R. Cruz. "Optimization and experiments: a survey." Applied Mechanics
Reviews 51.7 (1998): 435-448.
6. Rikards, R., et al. "Elaboration of optimal design models for composite materials from data of experiments." Mechanics
of composite materials 28.4 (1993): 295-304.
7. Annadurai, G., and R. Y. Sheeja. "Use of Box-Behnken design of experiments for the adsorption of verofix red using
biopolymer." Bioprocess engineering 18.6 (1998): 463-466.
8. Ferreira, SL Costa, et al. "Box-Behnken design: an alternative for the optimization of analytical methods." Analytica
chimica acta 597.2 (2007): 179-186.
9. Aslan, N., and Y. Cebeci. "Application of Box–Behnken design and response surface methodology for modeling of some
Turkish coals." Fuel 86.1-2 (2007): 90-97.

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