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…