Download - Orthogonal arrays-lecturer 8.pdf
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FURTHER ORTHOGONAL ARRAYSFURTHER 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 fewOrthogonal arrays - Matrix of numbers
each column – each factor or interactioneach 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-replicateof 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)
verypowerfularray
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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 T6CASE STUDY T6
A consumer magazine subscription service has four factors – A, B, C and D, each to be analysed at two levels. Also of interest are the interactions of BxC, BxD and CxD. Show the experimental design for this case
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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 interestfactor B,C,D – base factors + 2 x 2-way interactions
1=B2=C3=BC4=D5=BD6=CD7=A - fits!
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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 T7CASE STUDY T7
The rapid transport authority in a large metropolitan area has identified five factors, A, B, C, D and E, each to be investigated at 2 levels. Interactions AC and AD are also of interest. Determine an appropriate experimental design.
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Here – A, B, C, D, E + AC and AD
i.e. 7 factors/interactions (5+2) – candidate array = L8 (27)
currently – not an optionit 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 fitscompletely
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 ARRAYSi.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; B=C=D=E = 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 T8CASE STUDY T8A commercial bank has identified 5 factors (A-E) that have an impact on its volume of loans. There are 4 levels of factor A and 2 levels for each of the other factors. Determine an appropriate experimental design
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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, A=1 rows 3,4, A=2 rows 5,6, A=3 rows 7,8 A=4
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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 noisei.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.
standardprocedure
9 experimentsrun4 times
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2 extra columns…
Mean response = Ῡ mean of each set of 4 replicates
S/N ratio = Z – as given previous
Ῡ and Z – used in analysis phasei.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 1,2,3level 2 – experiments 4,5,6level 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:
3321
1yyyA ++
=
3321
1yyyA ++
=
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Type a:effect – not significanti.e. not worth bothering with (?)
Type b:effect = non-linearbest selection – region where curve is flattest (i.e. minimum gradienti.e. minimum variability with response variable
here – level 2 is the most robust setting
Type c: effect = linearhere – factor = adjustment parameter
gradient is constant ∴ constant variationbut can change mean response easily
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Can repeat the procedure with interaction effects:
Interaction of BxC…
and
( )
( )
1 2 7 81
3 4 5 62
4
4
y y y yBxC
y y y yBxC
+ + +=
+ + +=
CASE STUDY T9CASE STUDY T9
Various components of a drug for lung cancer have positive and negative effects depending on the amount used. Scientists have identified four independent factors that seem to affect the performance of the drug.
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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…367.1
35.24.20.4
367.03
2.65.94.4
867.13
8.13.75.3
3
2
1
−=−+−
=
−=−+−
=
=++−
=
A
A
A
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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 factorsi.e. set A and C to level 2
∴ optimal setting = A=B=C=D=2
NOTE/ not one of our original experiments!
This is the essence of Taguchi parameter design-- to find the best parameter settings using 2to find the best parameter settings using 2--stage stage optimization and indirect experimentationoptimization and indirect experimentation
of course – further testing will confirm this…