FURTHER ORTHOGONAL ARRAYS

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FURTHER ORTHOGONAL ARRAYS The Taguchi approach
to experimental design Create matrices from
factors and factor levels - either an inner
array or design matrix (controllable factors) -
or an outer array or noise matrix (uncontrollable
factors) Here number of experiments required
significantly reduced How? Orthogonal
arrays
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Orthogonal arrays - Trivial many versus vital
few Orthogonal arrays - Matrix of numbers each
column each factor or interaction each row
levels of factors and interactions Main
property every factor setting occurs same
number of times for every test setting of all
other factors Allows for lots of
comparisons Any two columns form a complete
2-factor factorial design Critical concept the
LINEAR GRAPH
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First example L4 array a half-replicate of a
23 experiment 4 experiments factors level 1
or 2 3 factors? look at the linear graph 2
nodes (columns 1 and 2) 1 linkage (between 1
and 2 i.e. 2 factors 1 interaction
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The L4 array cannot estimate 3 base factors (not
yet!) also the nodes are different designs -
associated with the degree of difficulty with
changing the level of a particular
factor Acknowledges that not all factors are
easy to change
Easy means easy to use as it only changes a
minimum number of times ? if one factor is harder
to change put it in column 1, as this only
changes once Same for any size array
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L8 (27) array here 7 factors at 2 levels or 7
entities at 2 levels IMPORTANT! there are no
3-way interactions or higher represented by this
method total replicate 128 tests here only
8!
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Also 2 linear graphs (templates for candidate
experiments) 4 main effects 3 interactions so
long as one of the graphs fits your experiment -
use the array! If not choose another or modify
the graph (later)
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Another template the L9 (34) array here 4
factors, each at 3 levels should be 81 tests -
actually 9 2 base factors only others
confounded with interactions
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Many others L16 (215) 5 base factors 10
2-way interactions
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L27 (313) very powerful array
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Also can have arrays for factors of varying
number of levels e.g. L18 (21 x 37) i.e. a
hybrid (see later)
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CASE STUDY T6
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7 factors/interactions 2 levels A,B,C,D and
BC, BD, CD ?27? check the linear graphs! if
they match use the L8 approach note factor A
stand-alone i.e. no interaction of
interest factor B,C,D base factors 2 x 2-way
interactions 1B 2C 3BC 4D 5BD 6CD 7A - fit
s!
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i.e. can we modify these graphs
(templates) to account for other experimental
designs? yes!
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CASE STUDY T7
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Here A, B, C, D, E AC and AD i.e. 7
factors/interactions (52) candidate array L8
(27) currently not an option it gives 4
factors 3 interactions we need 5 factors
2 interactions we can modify the graph
by breaking a link and creating a node from
it preliminary allocation interaction 6? (AE)
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Pull it out and turn it into a node! i.e. the
experiment now fits completely i.e. BUT/ factor
B and interaction AE are now confounded therefore
must assume AE insignificant Orthogonal
arrays lots of similar assumptions
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Orthogonal arrays - graphs can be used to see
what designs are possible either direct or
modified Assumes no higher order interactions
and that not all base factors or 2-way
interactions are necessary plus-side 128 tests
per replicate ? 8 tests!
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HYBRID ORTHOGONAL ARRAYS i.e. technique for when
not all factors have the same no. of
levels First find no. of degrees of
freedom for each factor (always 1 less than no.
of levels) i.e. A 3 BCDE 1 total
7 Same as for L8 array (7 columns) ? use L8 as
our hybrid design template each column a
2-level interaction ? 1 degree of freedom/column
CASE STUDY T8
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7 columns 3 for A and 1 each for the other
factors BUT/ which factor in which
column? Consult the linear graph must
identify a line that can be removed
easily e.g. remove 1,2 and 3 to
give
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? a new column 1 made up of old columns
1,2,3 ? 7 columns now 5 a new A
column sequentially index them i.e. rows 1,2,
A1 rows 3,4, A2 rows 5,6, A3 rows
7,8 A4
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Estimation of effects We have the experimental
design. now run it! (r times) How many
replicates? Often decided using noise
factors Why include noise factors? To identify
design factor levels that are least sensitive to
noise i.e. robust
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e.g. 4 factors A, B, C, D 3 noise factors E,
F, G need a design array (L9) and a noise array
(L4) i.e. standard procedure 9
experiments run 4 times
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2 extra columns Mean response ? mean of each
set of 4 replicates S/N ratio Z as given
previous ? and Z used in analysis phase i.e.
the parameter design phase Taguchi approach
uses simple plots to make inferences (ANOVA
also possible) Main effect of a factor factor A
levels 1,2,3 level 1 experiments
1,2,3 level 2 experiments 4,5,6 level
3 experiments 7,8,9
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? mean response when A is at level
1 etc Another example factor B at
level 3 For each factor 3 points now plot!!
3 types of plot
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Type a effect not significant i.e. not
worth bothering with (?) Type b effect
non-linear best selection region where curve
is flattest (i.e. minimum gradient i.e.
minimum variability with response
variable here level 2 is the most robust
setting Type c effect linear here factor
adjustment parameter gradient is constant ?
constant variation but can change mean response
easily
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Can repeat the procedure with interaction
effects Interaction of BxC and
CASE STUDY T9
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4 factors x 3 levels ? L9 (34) need to modify
the array ? assume no interaction factors Now
run tests (target value 0) Only 1
replicate ? no noise factors possible
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Main effects etc now plot
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So what? B and D non-linear A almost
linear C linear For a robust system set B
and D to level 2 to reduce variability Then move
the response value to zero using adjustment
factors i.e. set A and C to level 2 ? optimal
setting ABCD2 NOTE/ not one of our
original experiments! This is the essence of
Taguchi parameter design - to find the best
parameter settings using 2-stage optimization
and indirect experimentation of course further
testing will confirm this